int_{-frac{pi}{2}}^{frac{pi}{2}} log left(fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{2-\sin x}{2+\sin x}\right) d x$.Consider the function $f(x) = \log \left(\frac{2-\

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{2-\sin x}{2+\sin x}\right) d x$.Consider the function $f(x) = \log \left(\frac{2-\sin x}{2+\sin x}\right)$.Now,check if the function is even or odd by evaluating $f(-x)$:$f(-x) = \log \left(\frac{2-\sin(-x)}{2+\sin(-x)}\right) = \log \left(\frac{2+\sin x}{2-\sin x}\right)$.Since $\log \left(\frac{a}{b}\right) = -\log \left(\frac{b}{a}\right)$,we have:$f(-x) = -\log \left(\frac{2-\sin x}{2+\sin x}\right) = -f(x)$.Since $f(-x) = -f(x)$,the function $f(x)$ is an odd function.According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) d x = 0$.Therefore,$I = 0$.

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

Similar Questions

$\int_0^\pi x \sin x \cos^4 x \, dx = $

$\int_{0}^{1} \tan ^{-1}\left(\frac{2x-1}{1+x-x^{2}}\right) dx =$

By using the properties of definite integrals,evaluate the integral $\int_{2}^{8}|x-5| d x$.

$\int\limits_{\frac{1}{2}}^{3\frac{1}{2}} {\left\{ {\frac{1}{2}\,\left( {|x - 3| + |1 - x| - 4} \right)} \right\}\,dx} $ equals: Where $\{*\}$ denotes the fractional part function.

$\int_0^{\pi / 2} \frac{200 \sin x+100 \cos x}{\sin x+\cos x} d x$ is equal to (in $\pi$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module