If $f\left(x\right) = \frac{2- x\cos x}{2+x \cos x}$ and $ g\left(x\right) =\log_{e}x ., \left(x>0\right) $ then the value of integral $\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}} g\left(f\left(x\right)\right)dx $ is :
- $\log_e 3$
- $\log_e 2$
- $\log_e e$
- $\log_e 1$
The Correct Option is D
Solution and Explanation
$ \therefore I = \int^{\frac{\pi}{4}}_{-\frac{\pi}{4}} \ell n\left(\frac{2-x.\cos x}{2+x\cos x}\right)dx $
$ = \int^{\frac{\pi}{4}}_{0} \left(\ell n \left(\frac{2-x\cos x}{2+x.\cos x}\right) + \ell n\left(\frac{2+x.\cos x}{2-x.\cos x}\right)\right)dx $
$ = \int^{\frac{\pi}{2}}_{0} \left(0\right)dx =0 =\log_{e} \left(1\right) $
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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.
