Question:
The value of the integral $ \int ^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \bigg ( x^2 + \log \frac{\pi - x }{ \pi + x } \bigg ) \, \cos \, x \, dx $
The value of the integral $ \int ^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \bigg ( x^2 + \log \frac{\pi - x }{ \pi + x } \bigg ) \, \cos \, x \, dx $
Updated On: Jun 14, 2022
- $0$
- $\frac{\pi^2 }{2} - 4 $
- $\frac{\pi^2 }{2} + 4 $
- $\frac{ \pi ^2 }{ 2 }$
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The Correct Option is B
Solution and Explanation
$ I = \int ^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \bigg [ x^2 + \log \frac{\pi - x }{ \pi + x } \bigg ] \, \cos \, x \, dx $
$As, \int^a_{-a} f ( x ) \, dx = 0 , when \, f ( -x) = -f ( x )$
$\therefore I = I = \int ^{\frac{\pi}{2}}_{-\frac{\pi}{2}} x^2 \, \cos \, x \, dx + 0 = 2 \int^{\frac{\pi}{2} }_0 ( x^2 \cos x ) \, dx $
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 2 \bigg \{ ( x^2 \, \sin \, x )^{\frac{\pi}{2}}_0 - \int^{\frac{\pi}{2} }_0 \, 2 \, x.\sin\, x \, dx \bigg \}$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 2 \bigg [ \frac{\pi^2 }{4} -2 \bigg \{ ( - x .\cos \, x )^{\frac{\pi}{2}} _0 - \int^{\frac{\pi}{2}}_0 1 . ( - \cos \, x ) \, dx \bigg \} \bigg ]$
$ 2 \bigg [ \frac{ \pi^2}{4} - 2 ( \sin \, x )^{\frac{\pi}{2}}_0\bigg ] = 2 \bigg [ \frac{\pi^2}{4} - 2 \bigg ] = \bigg ( \frac{ \pi^2 }{ 2} - 4\bigg )$
$As, \int^a_{-a} f ( x ) \, dx = 0 , when \, f ( -x) = -f ( x )$
$\therefore I = I = \int ^{\frac{\pi}{2}}_{-\frac{\pi}{2}} x^2 \, \cos \, x \, dx + 0 = 2 \int^{\frac{\pi}{2} }_0 ( x^2 \cos x ) \, dx $
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 2 \bigg \{ ( x^2 \, \sin \, x )^{\frac{\pi}{2}}_0 - \int^{\frac{\pi}{2} }_0 \, 2 \, x.\sin\, x \, dx \bigg \}$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 2 \bigg [ \frac{\pi^2 }{4} -2 \bigg \{ ( - x .\cos \, x )^{\frac{\pi}{2}} _0 - \int^{\frac{\pi}{2}}_0 1 . ( - \cos \, x ) \, dx \bigg \} \bigg ]$
$ 2 \bigg [ \frac{ \pi^2}{4} - 2 ( \sin \, x )^{\frac{\pi}{2}}_0\bigg ] = 2 \bigg [ \frac{\pi^2}{4} - 2 \bigg ] = \bigg ( \frac{ \pi^2 }{ 2} - 4\bigg )$
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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.
