The value of $\int^{\pi}_{-\pi} \frac{ \cos^2 \, x }{ 1 + a^x } \, dx , a> 0 $ is
- $ \pi $
- $ a \, \pi $
- $\frac{\pi}{2} $
- $ 2 \pi $
The Correct Option is C
Solution and Explanation
Let \(I = \int^{\pi}_{-\pi} \frac{ \cos^2 \, x }{ 1 + a^x } \, dx ...................(1)\)
\(= \int^{\pi}_{-\pi} \frac{ \cos^2 \, ( -x ) }{ 1 + a^{-x} } \, dx\)
\(I = \int^{\pi}_{-\pi} \frac{ \cos^2 \, {x } }{ 1 + a^{-x} } \, dx................(2)\)
On adding Eqs. (i) and (ii), we get
\(2I = \int^{\pi} _{-\pi} \bigg ( \frac{ 1 + a^x}{1 + a^x } \bigg ) \cos^2 \, x \, d \, x\)
\(= \int^{\pi} _{-\pi} \cos^2 \, x \, d \, x = 2 \int^{\pi}_{0} \frac{ 1 + \cos \, 2x}{2} dx\)
\(= [x]^{\pi}_{0} + 2 \int^{\pi/ 2 }_ 0 \, \cos \, 2x \, dx = \pi + 0\)
\(2I = \pi \)
\(\Rightarrow I =\frac{\pi} {2}\)
So. the correct answer is (C): \(\frac {\pi}{2}\)
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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.
