int_0^{pi /2} frac{cos x}{(1 + sin x)(2 + sin | vedclass

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

Question

(A) Let $I = \int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. When $x = 0$,$t = 0$. Whe

Vedclass · Accepted Answer

$\log \frac{4}{3}$

Vedclass · Answer

(A) Let $I = \int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. When $x = 0$,$t = 0$. When $x = \pi/2$,$t = 1$. The integral becomes $I = \int_0^1 \frac{1}{(1 + t)(2 + t)} \,dt$. Using partial fractions,$\frac{1}{(1 + t)(2 + t)} = \frac{1}{1 + t} - \frac{1}{2 + t}$. Thus,$I = \int_0^1 \left( \frac{1}{1 + t} - \frac{1}{2 + t} \right) \,dt$. $I = [\log |1 + t| - \log |2 + t|]_0^1$. $I = [\log \frac{1 + t}{2 + t}]_0^1$. $I = \log \frac{2}{3} - \log \frac{1}{2} = \log \left( \frac{2/3}{1/2} \right) = \log \frac{4}{3}$.

$\int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx = $

Similar Questions

If $f(x) = \int_{-1}^{x} |t| dt$,then for any $x \geq 0$,$f(x)$ is equal to

If $[x]$ is the greatest integer function not greater than $x$,then $\int_{0}^{11} [x] dx$ is equal to

$\int_0^\pi \frac{1}{1+\sin x} dx$ is equal to

$\int_0^1 {{\tan ^{ - 1}}x\,dx = } $

$\int_0^3 |x^2 - 3x + 2| dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module