int frac{cos x}{(1 + sin x)(2 + sin x)} ,dx = | vedclass

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

Question

(C) Let $I = \int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. The integral becomes $I = \int \frac

Vedclass · Accepted Answer

$\log \left| \frac{1 + \sin x}{2 + \sin x} \right| + c$

Vedclass · Answer

(C) Let $I = \int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. The integral becomes $I = \int \frac{dt}{(t + 1)(t + 2)}$. Using partial fractions,$\frac{1}{(t + 1)(t + 2)} = \frac{1}{t + 1} - \frac{1}{t + 2}$. Integrating both terms,$I = \int \frac{1}{t + 1} \,dt - \int \frac{1}{t + 2} \,dt = \log |t + 1| - \log |t + 2| + c$. Using the property $\log a - \log b = \log \frac{a}{b}$,we get $I = \log \left| \frac{t + 1}{t + 2} \right| + c$. Substituting $t = \sin x$ back,$I = \log \left| \frac{1 + \sin x}{2 + \sin x} \right| + c$.

$\int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx = $

Similar Questions

If $\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+b \tan ^{-1} x+\frac{1}{5} \log |x+2|+c$,then

$\frac{x^4}{(x^2+1)(x^2+3)} =$

Integrate the function: $\frac{1}{(x^{2}+1)(x^{2}+4)}$

$\int \frac{dx}{\sin x + \sin 2x} = $

If $\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k} \log_e \left\{ \frac{|x-3|^3|x+1|}{(x-2)^4} \right\}+c$, then the value of $k$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module