If int_0^{frac{pi}{2}} log cos x , dx = frac{ | vedclass

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

Question

(D) We are given that $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$.We need to evaluate $I = \int_0^{\frac{

Vedclass · Accepted Answer

$\frac{\pi}{2} \log 2$

Vedclass · Answer

(D) We are given that $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$.We need to evaluate $I = \int_0^{\frac{\pi}{2}} \log \sec x \, dx$.Since $\sec x = \frac{1}{\cos x}$,we have $\log \sec x = \log \left(\frac{1}{\cos x}\right) = \log 1 - \log \cos x = -\log \cos x$.Therefore,$I = \int_0^{\frac{\pi}{2}} -\log \cos x \, dx$.$I = -\int_0^{\frac{\pi}{2}} \log \cos x \, dx$.Substituting the given value,$I = -\left[\frac{\pi}{2} \log \left(\frac{1}{2}\right)\right]$.Since $\log \left(\frac{1}{2}\right) = \log (2^{-1}) = -\log 2$,we get $I = -\left[\frac{\pi}{2} (-\log 2)\right] = \frac{\pi}{2} \log 2$.

If $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$,then $\int_0^{\frac{\pi}{2}} \log \sec x \, dx = $

Similar Questions

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

$ \int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x = $

By using the properties of definite integrals,evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x}{\sin ^{5} x+\cos ^{5} x} d x$.

Let $g(t) = \int_{-\pi/2}^{\pi/2} \cos \left(\frac{\pi}{4} t + f(x)\right) \, dx$,where $f(x) = \log_e \left(x + \sqrt{x^2 + 1}\right)$,$x \in R$. Then which one of the following is correct?

By using the properties of definite integrals,evaluate the integral $\int_{0}^{\pi} \frac{x \, dx}{1+\sin x}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module