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Application of Even-Odd Properties in Definite Integrals

Application of Even-Odd Properties in Definite Integrals

Edited By Komal Miglani | Updated on Jul 02, 2025 07:59 PM IST

Integration is one of the important parts of Calculus, which applies to measuring the change in the function at a certain point. Mathematically, it forms a powerful tool by which slopes of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These integration concepts have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

Application of Even-Odd Properties in Definite Integrals
Application of Even-Odd Properties in Definite Integrals

Application of Even-Odd Properties in Definite Integration

Definite integration calculates the area under a curve between two specific points on the x-axis.

Let f be a function of x defined on the closed interval [a, b]. F be another function such that $\frac{d}{d x}(F(x))=f(x)$ for all x in the domain of f, then $\int_a^b f(x) d x=[F(x)+c]_a^b=F(b)-F(a)$is called the definite integral of the function f(x) over the interval [a, b], where a is called the lower limit of the integral and b is called the upper limit of the integral.

Definite integrals have properties that relate to the limits of integration.

Property 1

$\int_{-a}^a f(x) d x=\left\{\begin{array}{ccc}0, & \text { if } f \text { is an odd function } & \text { i.e. } f(-x)=-f(x) \\ 2 \int_0^a f(x) d x, & \text { if } f \text { is an even function } & \text { i.e. } f(-x)=f(x)\end{array}\right.$

Proof:

$\begin{aligned} \int_{-\mathrm{a}}^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx} & =\underbrace{\int_{-a}^0 f(x) d x}_{x=-t}+\int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \\ & =\int_a^0 f(-t)(-d t)+\int_0^a f(x) d x \\ & =\int_0^a f(-x)(d x)+\int_0^a f(x) d x \\ & =\left\{\begin{array}{cc}-\int_0^a f(x) d x+\int_0^a f(x) d x, & \text { if } \mathrm{f}(\mathrm{x}) \text { is odd } \\ \int_0^a f(x) d x+\int_0^a f(x) d x, & \text { if } \mathrm{f}(\mathrm{x}) \text { is even }\end{array}\right. \\ & =\left\{\begin{array}{cc}0, & \text { if } f \text { is an odd function } \\ 2 \int_0^a f(x) d x, & \text { if } f \text { is an even function }\end{array}\right.\end{aligned}$

Proof using Graph

The graph of the odd function is symmetric about the origin, as shown in the above figure

So, if $\int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\alpha$ then, $\int_{-\mathrm{a}}^0 \mathrm{f}(\mathrm{x}) \mathrm{dx}=-\alpha$

$
\therefore \quad \int_{-a}^a f(x) d x=0
$


The graph of the even function is symmetric about the y-axis, as shown in the above figure

$\begin{array}{ll}\text { So, } & \int_{-a}^0 \mathrm{f}(\mathrm{x}) \mathrm{dx}=\int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\alpha \\ \therefore & \int_{-a}^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=2 \alpha=2 \int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\end{array}$

Corollary:

$\int_0^{2 a} f(x) d x=\left\{\begin{array}{cc}2 \int_0^a f(x) d x, & \text { if } f(2 a-x)=f(x) \\ 0, & \text { if } f(2 a-x)=-f(x\end{array}\right.$

Property 9

If f(x) is a periodic function with period T, then the area under f(x) for n periods would be n times the area under f(x) for one period, i.e.

$\int_0^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\mathrm{n} \int_0^{\mathrm{T}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$

Proof:

Graphical Method

f(x) is a periodic function with period T. Consider the following graph of function f(x).

The graph of the function is the same in each of the interval (0, T), (T, 2T), (2T, 3T) ……..

So,

$\begin{aligned} \int_0^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{d} & =\text { total shaded area } \\ & =\mathrm{n} \times(\text { area in the interval }(0, \mathrm{~T})) \\ & =\mathrm{n} \int_0^{\mathrm{T}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\end{aligned}$

Property 10

$\int_a^{a+n T} f(x) d x=\int_0^{a T} f(x) d x=n \int_0^T f(x) d x$

Proof:

$\begin{aligned} & \text { Let, } \begin{aligned} & \mathrm{I}=\int_{\mathrm{a}}^{\mathrm{a}+\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \\ &=\int_{\mathrm{a}}^0 \mathrm{f}(\mathrm{x}) \mathrm{dx}+\int_0^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx}+\underbrace{\int_{n T}^{a+n T} f(x) d x}_{x=y+n T} \\ & \Rightarrow \mathrm{dx}=\mathrm{dy} \text { and when } \mathrm{x}=\mathrm{nT} \text { then } \mathrm{y}=0 \text { and } \mathrm{x}=\mathrm{a}+\mathrm{nT}, \mathrm{y}=\mathrm{a} \\ &=\int_{\mathrm{a}}^0 \mathrm{f}(\mathrm{x}) \mathrm{dx}+\int_0^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx}+\int_0^{\mathrm{a}} \mathrm{f}(\mathrm{y}) \mathrm{dy} \\ &=\mathrm{n} \int_0^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \\ & {\left[\because \int_0^{\mathrm{a}} \mathrm{f}(\mathrm{y}) \mathrm{dy}=\int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \text { and } \int_0^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=-\int_{\mathrm{a}}^0 \mathrm{f}(\mathrm{x}) \mathrm{dx}\right] }\end{aligned}\end{aligned}$

Property 11

$
\int_{\mathrm{a}+\mathrm{nT}}^{\mathrm{b}+\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}
$


Property 12

$
\int_{\mathrm{mT}}^{\mathrm{nT}} f(x) d x=(n-m) \int_0^{\mathrm{T}} f(x) d x
$

Where ‘T’ is the period and m and n are Integers

Recommended Video Based on Application of Even-Odd Properties in Definite Integration

Solved Example Based on Application of Even-Odd Properties in Definite Integration

Example 1: $\int_{-3 \pi / 2}^{-\pi / 2}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x$ is equal to

1) $\frac{\pi^4}{32}$
2) $\frac{\pi^4}{32}+\frac{\pi}{2}$
3) $\frac{1}{2}$
4) $\frac{\pi}{2}-1$

Solution

As learnt in concept

Properties of definite integration -

If $f(x)$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^a f(x) d x=2\left\{\int_o^a f(x) d x\right\}$

wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.


$
I=\int_{\frac{-3 \pi}{2}}^{-\frac{\pi}{2}}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x
$


Put $x+\pi=t$

$
\begin{aligned}
& I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[t^3+\cos ^2 t\right] d t \\
& I=2 \int_0^{\frac{\pi}{2}} \cos ^2 t d \\
& =\int_0^{\frac{\pi}{2}}[1+\operatorname{Cos} 2 t] d t \\
& {[t]_0^{\frac{\pi}{2}}+\left[\frac{\operatorname{Sin} 2 t}{2}\right]_0^{\frac{\pi}{2}}} \\
& \frac{\pi}{2}+0
\end{aligned}
$

Example 2: $\int_{-\pi}^\pi \frac{2 x(1+\sin x)}{1+\cos ^2 x} d x$ is

1) $\pi^2 / 4$
2) $\pi^2$
3) 0
4) $\pi / 2$

Solution

As we learnt in

Properties of definite integration -

If $f(x)$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^a f(x) d x=2\left\{\int_o^a f(x) d x\right\}$

- wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.

And,

Properties of Definite Integration -

If $f(x)$ is an odd function of $x$ then integral of the function from -a to a is ZERO

$\int_{-a}^a f(x) d x=0$
- wherein

Check

Odd function $f(-x)=-f(x)$

And,

Properties of Definite Integration -

$\int_0^{2 a} f(x) d x=\int_0^a[f(x)+f(-x)] d x$

$=\left\{\begin{array}{cc}2 \int_0^a f(x) d x & \text { if } f(2 a-x)=f(x) \\ 0 & \text { if } f(2 a-x)=-f(x)\end{array}\right.$

- $I=\int_{-\pi}^\pi \frac{2 x(1+\sin x) d x}{1+\cos ^2 x}$

$I=\int_0^\pi\left(\frac{2 x(1+\sin x)-2 x+2 x \sin x}{\left(1+\cos ^2 x\right)}\right) d x$

$\begin{aligned} & I=\int_0^\pi \frac{4 x \sin x}{\left(1+\cos ^2 x\right)} d x \\ & I=\int_0^{\frac{\pi}{2}} \frac{4 \sin x}{1+\cos ^2 x}(x+\pi-x) d x \\ & I=4 \pi \int_0^{\frac{\pi}{2}} \frac{\sin x d x}{1+\cos ^2 x} \\ & I=4 \pi\left[\tan ^{-1}(\cos x)\right]^\pi \\ & I=4 \pi\left[0+\frac{\pi}{4}\right]=\pi^2\end{aligned}$

Example 3: The value of the integral $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^4 x\left[1+\log \left[\frac{2+\sin x}{2-\sin x}\right]\right] d x$ is :

1) 0
2) $\frac{3}{4}$
3) $\frac{3}{8} \pi$
4) $\frac{3}{16} \pi$

Solution

As we learned

Properties of definite integration -

If $f(x)$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^a f(x) d x=2\left\{\int_0^a f(x) d x\right\}$

- wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.

Check

Odd function $f(-x)=-f(x)$

$\int_{-\pi}^{\frac{1}{2}} \sin ^4 x\left(1+\log \left(\frac{2+\sin x}{2-\sin x}\right)\right) d x$

$=2 \int_0^{\frac{\pi}{2}} \sin ^4 x d x+2 \int_0^{\frac{x}{2}} \sin ^4 x \log \frac{2+\sin x}{2-\sin x} d x$

Since value of $\int_0^{\frac{5}{2}} \sin ^4 x \log \frac{2+\sin x}{2-\sin x} d x=0$ (Odd function)

So,

$\begin{aligned} & \sin ^4 x=\frac{1}{4}(1-\cos 2 x)^2 \\ & \sin ^4 x=\frac{1}{4}\left(1+\cos ^2 2 x-2 \cos 2 x\right) \\ & \sin ^4 x=\frac{1}{4}\left(1+\frac{1+\cos 4 x}{2}-2 \cos 2 x\right)\end{aligned}$

$\begin{aligned} & I=2 \cdot \int_0^{\frac{\pi}{2}} \sin ^4(x) d x \\ & I=2 \cdot \int_0^{\frac{\pi}{2}} \frac{1}{4}\left(\frac{3+\cos (4 x)-8 \cos (2 x)}{2}\right) d x \\ & =\frac{3 \pi}{8}\end{aligned}$

Example 4: $\int_{-\Pi / 2}^{\Pi / 2} \frac{\cos x}{1+e^x} d x$ is equal to?

1) $1$

2) $2 $

3) $-1$

4) $-2$

Solution

As we learnt

Properties of definite integration -

If $f(x)$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^a f(x) d x=2\left\{\int_0^a f(x) d x\right\}$

- wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.


$\begin{aligned} & I=\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x=\int_0^{\pi / 2}\left(\frac{\cos x}{1+e^x}+\frac{\cos (-x)}{1+e^x}\right) d x=\int_0^{\pi / 2} \cos x d x= \\ & {[\sin x]_0^{\pi / 2}=1}\end{aligned}$

Example 5: $\int_{-1}^1 \frac{\log \left(x+\sqrt{1+x^2}\right)}{x+\log \left(x+\sqrt{1+x^2}\right)}(f(x)-f(-x)) d x$ is euqal to?

1) $0$

2) $2 \int_0^1 \frac{\log \left(x+\sqrt{1+x^2}\right)}{x+\log \left(x+\sqrt{1+x^2}\right)}(f(x)-f(-x)) d x$

3) $2 f(x)$

4) none of these

Solution

As we learnt

Properties of definite integration -

If $f(x)$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^a f(x) d x=2\left\{\int_o^a f(x) d x\right\}$

- wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.

$\operatorname{Let} \phi(x)=\frac{\log \left(x+\sqrt{1+x^2}\right)}{x+\log \left(x+\sqrt{1+x^2}\right)} ; g(x)=(f(x)-f(-x))$

f(x) is an even function $\int_{-1}^1(\phi(x) \cdot g(x)) d x=0$

g(x) is an odd function.

Summary

Definite integration is a powerful tool in calculus that allows us to calculate the area under a curve between two specific points. Even-odd integration is useful when the function is symmetric around the x-axis or y-axis, It provides a deeper understanding of mathematical ideas paramount for later developments in many scientific and engineering disciplines.

Frequently Asked Questions (FAQs)

1. How does the even-odd property relate to the concept of symmetry in calculus?
The even-odd property is a direct application of symmetry in calculus. It shows how the symmetry of a function about the y-axis (even) or origin (odd) affects the evaluation of definite integrals, allowing for simplification and sometimes immediate conclusions without calculation.
2. What happens when you integrate an even function from -a to a?
When you integrate an even function from -a to a, the result is twice the integral from 0 to a. This is because the function is symmetric about the y-axis, so the areas on both sides are equal.
3. How can you use the even-odd property to evaluate ∫[-π, π] sin(x) dx without calculation?
Since sin(x) is an odd function, its integral from -π to π is zero. We can conclude this without calculation because the areas above and below the x-axis cancel out due to the function's symmetry about the origin.
4. If f(x) is even and g(x) is odd, what can you say about the integral of their product from -a to a?
The integral of the product of an even function f(x) and an odd function g(x) from -a to a is always zero. This is because the resulting function is odd, and the integral of an odd function over a symmetric interval is zero.
5. Can you apply the even-odd property to improper integrals?
Yes, the even-odd property can be applied to improper integrals over symmetric intervals. For example, ∫[-∞, ∞] f(x) dx = 2∫[0, ∞] f(x) dx if f(x) is even, and it equals zero if f(x) is odd, provided the integrals converge.
6. What is the even-odd property in definite integrals?
The even-odd property in definite integrals refers to the symmetry of functions and how it affects the integral's value. For even functions, the integral from -a to a is twice the integral from 0 to a. For odd functions, the integral from -a to a is zero.
7. Why does the integral of an odd function from -a to a equal zero?
The integral of an odd function from -a to a is zero because the areas above and below the x-axis are equal in magnitude but opposite in sign, canceling each other out due to the function's symmetry about the origin.
8. How does the even-odd property simplify integration?
The even-odd property simplifies integration by allowing us to reduce the interval of integration. For even functions, we can integrate from 0 to a and double the result, while for odd functions, we know the integral from -a to a is zero without calculation.
9. How can you identify if a function is even or odd?
An even function is symmetric about the y-axis, meaning f(-x) = f(x). An odd function is symmetric about the origin, meaning f(-x) = -f(x). You can test this by replacing x with -x in the function and seeing if it equals f(x) or -f(x).
10. How does the even-odd property apply to definite integrals with limits other than -a to a?
The even-odd property is most useful for integrals from -a to a, but it can be applied to other intervals by shifting the function. For example, ∫[b-a, b+a] f(x-b) dx can be transformed to ∫[-a, a] f(u) du by substituting u = x-b.
11. Can a function be both even and odd?
A function can be both even and odd only if it is identically zero for all x. This is because the only function that satisfies both f(-x) = f(x) and f(-x) = -f(x) is f(x) = 0.
12. What happens to the even-odd property when you compose functions?
When composing functions, the even-odd property follows these rules:
13. How does the even-odd property relate to the concept of parity in physics?
The even-odd property in mathematics is analogous to the concept of parity in physics. Even functions correspond to even parity (symmetric under reflection), while odd functions correspond to odd parity (anti-symmetric under reflection). This connection helps in understanding physical symmetries and conservation laws.
14. What is the significance of the even-odd property in solving differential equations?
The even-odd property is useful in solving differential equations, especially when looking for particular solutions. If the forcing function in a differential equation is even or odd, it can suggest the form of the particular solution and simplify the integration process.
15. How can you use the even-odd property to simplify the calculation of moments in probability theory?
In probability theory, the even-odd property can simplify the calculation of moments. For symmetric distributions (which are even functions), all odd moments about the mean are zero. This property allows us to quickly determine certain characteristics of probability distributions without extensive calculations.
16. What is the effect of a linear transformation on the even-odd property of a function?
Linear transformations can change the even-odd property of a function:
17. How does the even-odd property relate to the concept of symmetric and antisymmetric matrices?
The even-odd property in functions is analogous to symmetric and antisymmetric matrices in linear algebra. Even functions correspond to symmetric matrices (A = A^T), while odd functions correspond to antisymmetric matrices (A = -A^T). This connection helps in understanding properties of matrices and their eigenvalues.
18. How does the even-odd property affect the Taylor series expansion of a function?
The even-odd property affects the Taylor series expansion of a function:
19. What is the relationship between the even-odd property and the concept of odd and even functions in complex analysis?
In complex analysis, the concepts of even and odd functions are extended to complex variables. An even function f(z) satisfies f(-z) = f(z), while an odd function satisfies f(-z) = -f(z) for all complex z. These properties affect contour integrals and residue calculations in ways similar to real-valued functions.
20. What is the effect of composition with absolute value on the even-odd property?
Composition with absolute value affects the even-odd property as follows:
21. How does the even-odd property relate to the concept of odd and even permutations in group theory?
While not directly related, the even-odd property in calculus and the concept of odd and even permutations in group theory share the idea of classifying mathematical objects into two categories based on their behavior under certain operations. This parallel can help students understand the broader application of parity concepts in mathematics.
22. How does the even-odd property affect the behavior of a function's derivative?
The even-odd property affects derivatives as follows:
23. What is the significance of the even-odd property in Fourier transforms?
In Fourier transforms, the even-odd property is crucial:
24. How can you use the even-odd property to determine the nature of critical points in a function?
The even-odd property can help determine the nature of critical points:
25. What is the relationship between the even-odd property and the concept of symmetry in physics problems?
The even-odd property in mathematics directly relates to symmetry in physics problems. Many physical systems exhibit symmetries that can be described mathematically as even or odd functions. Recognizing these symmetries can simplify problem-solving, reduce computational complexity, and provide insights into the system's behavior.
26. How does the even-odd property relate to the concept of symmetric and antisymmetric wavefunctions in quantum mechanics?
In quantum mechanics, wavefunctions can be classified as symmetric or antisymmetric, which is analogous to even and odd functions in calculus. Symmetric wavefunctions (even) remain unchanged under particle exchange, while antisymmetric wavefunctions (odd) change sign. This property is crucial in determining the behavior of quantum systems and particle statistics.
27. What is the relationship between the even-odd property and the concept of parity conservation in particle physics?
The even-odd property in mathematics is analogous to parity conservation in particle physics. Parity transformation in physics is equivalent to spatial inversion (x → -x), which is related to the even-odd property of functions. Understanding this connection helps in analyzing physical systems
28. How does knowing a function is even or odd help in evaluating its Fourier series?
Knowing a function is even or odd simplifies its Fourier series. Even functions have only cosine terms in their Fourier series, while odd functions have only sine terms. This reduces the number of coefficients to calculate and simplifies the integration process.
29. What is the relationship between the even-odd property and the area under a curve?
The even-odd property directly relates to the area under a curve. For even functions, the area from -a to 0 equals the area from 0 to a, so the total area is twice that of 0 to a. For odd functions, the areas above and below the x-axis cancel out, resulting in a net area of zero from -a to a.
30. How can you use the even-odd property to check your integration results?
You can use the even-odd property as a quick check for your integration results. If you're integrating an even function from -a to a, your result should be an even number times the integral from 0 to a. If you're integrating an odd function from -a to a, your result should be zero.
31. Can you use the even-odd property for functions defined piecewise?
Yes, you can use the even-odd property for piecewise functions, but you must ensure the entire function is even or odd. Check the symmetry of each piece and how they connect at the boundaries. If the overall function maintains the required symmetry, you can apply the property.
32. How does the even-odd property affect the mean value of a function over a symmetric interval?
For an odd function, the mean value over a symmetric interval [-a, a] is always zero because the integral is zero. For an even function, the mean value over [-a, a] is the same as the mean value over [0, a] because the function is symmetric about the y-axis.
33. How does the even-odd property relate to the concept of odd and even extensions of functions?
The even-odd property is closely related to odd and even extensions of functions. When we extend a function defined on [0, a] to [-a, a], we can create an even extension (mirroring the function) or an odd extension (mirroring and negating). These extensions allow us to apply the even-odd property to functions originally defined only on positive intervals.
34. Can you apply the even-odd property to multivariable integrals?
Yes, the even-odd property can be extended to multivariable integrals. For functions of two variables, f(x,y), we can have evenness or oddness in each variable separately. This leads to symmetries in double integrals over rectangular regions symmetric about both axes.
35. How does the even-odd property affect the convergence of improper integrals?
The even-odd property can affect the convergence of improper integrals. For example, if f(x) is an odd function, ∫[-∞, ∞] f(x) dx may converge even if ∫[0, ∞] f(x) dx diverges, because the positive and negative infinities can cancel each other out.
36. What is the relationship between the even-odd property and the symmetry of graphs?
The even-odd property is directly related to the symmetry of graphs. Even functions have graphs symmetric about the y-axis, while odd functions have graphs symmetric about the origin. This visual representation helps in understanding and applying the property to definite integrals.
37. Can you use the even-odd property to simplify the evaluation of definite integrals involving trigonometric functions?
Yes, the even-odd property is particularly useful for integrals involving trigonometric functions. Since sin(x) and tan(x) are odd functions, while cos(x) is even, you can often simplify or immediately evaluate integrals of these functions over symmetric intervals.
38. How can you use the even-odd property to simplify the evaluation of definite integrals involving absolute value functions?
The even-odd property is useful for integrals involving absolute value functions because |x| is an even function. This means that ∫[-a, a] |f(x)| dx = 2∫[0, a] |f(x)| dx, which can simplify calculations, especially when f(x) itself has known symmetry properties.
39. Can you use the even-odd property to simplify the evaluation of definite integrals involving polynomial functions?
Yes, the even-odd property is useful for integrals of polynomial functions. You can separate a polynomial into its even and odd parts, then use the property to simplify the integration. For example, in ∫[-a, a] (x^3 + x^2 + x + 1) dx, the x^3 and x terms (odd) integrate to zero, while the x^2 and 1 terms (even) can be simplified using the property.
40. How does the even-odd property affect the evaluation of definite integrals involving exponential functions?
The even-odd property affects integrals of exponential functions as follows:
41. Can you use the even-odd property to simplify the evaluation of definite integrals involving rational functions?
Yes, the even-odd property can be applied to rational functions. You can decompose a rational function into its even and odd parts, then apply the property to each part separately. This is particularly useful for integrals of the form ∫[-a, a] P(x)/Q(x) dx, where P(x) and Q(x) are polynomials.
42. What is the effect of definite integration on the even-odd property of a function?
Definite integration can change the even-odd property of a function:
43. How can you use the even-odd property to simplify the evaluation of improper integrals involving oscillating functions?
The even-odd property is particularly useful for improper integrals of oscillating functions. For example, in integrals like ∫[0, ∞] sin(x)/x dx, you can extend the interval to [-∞, ∞] and use the oddness of sin(x)/x to simplify the problem. This technique is often used in Fourier analysis and signal processing.

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Arrange the following Cobalt complexes in the order of incresing Crystal Field Stabilization Energy (CFSE) value. Complexes :  

\mathrm{\underset{\textbf{A}}{\left [ CoF_{6} \right ]^{3-}},\underset{\textbf{B}}{\left [ Co\left ( H_{2}O \right )_{6} \right ]^{2+}},\underset{\textbf{C}}{\left [ Co\left ( NH_{3} \right )_{6} \right ]^{3+}}\: and\: \ \underset{\textbf{D}}{\left [ Co\left ( en \right )_{3} \right ]^{3+}}}

Choose the correct option :
Option: 1 \mathrm{B< C< D< A}
Option: 2 \mathrm{B< A< C< D}
Option: 3 \mathrm{A< B< C< D}
Option: 4 \mathrm{C< D< B< A}

The type of hybridisation and magnetic property of the complex \left[\mathrm{MnCl}_{6}\right]^{3-}, respectively, are :
Option: 1 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 2 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 3 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 4 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 5 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 6 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 7 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 8 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 9 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 10 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 11 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 12 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 13 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 14 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 15 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 16 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
The number of geometrical isomers found in the metal complexes \mathrm{\left[ PtCl _{2}\left( NH _{3}\right)_{2}\right],\left[ Ni ( CO )_{4}\right], \left[ Ru \left( H _{2} O \right)_{3} Cl _{3}\right] \text { and }\left[ CoCl _{2}\left( NH _{3}\right)_{4}\right]^{+}} respectively, are :
Option: 1 1,1,1,1
Option: 2 1,1,1,1
Option: 3 1,1,1,1
Option: 4 1,1,1,1
Option: 5 2,1,2,2
Option: 6 2,1,2,2
Option: 7 2,1,2,2
Option: 8 2,1,2,2
Option: 9 2,0,2,2
Option: 10 2,0,2,2
Option: 11 2,0,2,2
Option: 12 2,0,2,2
Option: 13 2,1,2,1
Option: 14 2,1,2,1
Option: 15 2,1,2,1
Option: 16 2,1,2,1
Spin only magnetic moment of an octahedral complex of \mathrm{Fe}^{2+} in the presence of a strong field ligand in BM is :
Option: 1 4.89
Option: 2 4.89
Option: 3 4.89
Option: 4 4.89
Option: 5 2.82
Option: 6 2.82
Option: 7 2.82
Option: 8 2.82
Option: 9 0
Option: 10 0
Option: 11 0
Option: 12 0
Option: 13 3.46
Option: 14 3.46
Option: 15 3.46
Option: 16 3.46

3 moles of metal complex with formula \mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{3} gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of CO in the complex is_______.
(Round off to the nearest integer)
 

The overall stability constant of the complex ion \mathrm{\left [ Cu\left ( NH_{3} \right )_{4} \right ]^{2+}} is 2.1\times 10^{1 3}. The overall dissociation constant is y\times 10^{-14}. Then y is ___________(Nearest integer)
 

Identify the correct order of solubility in aqueous medium:

Option: 1

Na2S > ZnS > CuS


Option: 2

CuS > ZnS > Na2S


Option: 3

ZnS > Na2S > CuS


Option: 4

Na2S > CuS > ZnS


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