Question:
Prove that the value of the integral, $\int\limits_0^{2a} [f(x)/ f(x) + f(2a-x)] dx$ is equal to a
Prove that the value of the integral, $\int\limits_0^{2a} [f(x)/ f(x) + f(2a-x)] dx$ is equal to a
Updated On: Jun 14, 2022
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The Correct Option is A
Solution and Explanation
Let $ I=\int \limits_0^{2a} \frac{f(x)}{ f(x )+f(2a -x )} dx $ .......(i)
$ I=\int \limits_0^{2a} \frac{f( 2a-x)}{ f(x )+f(2a -x )} dx $ ........(ii)
On adding Eqs. (i) and (ii), we get
$2I= \int \limits_0^{2a} 1 dx = 2a \Rightarrow I=a $
$ I=\int \limits_0^{2a} \frac{f( 2a-x)}{ f(x )+f(2a -x )} dx $ ........(ii)
On adding Eqs. (i) and (ii), we get
$2I= \int \limits_0^{2a} 1 dx = 2a \Rightarrow I=a $
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
