int left( frac{x}{x cos x - sin x} right)^2 d | vedclass

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

Question

(B) Let $I = \int \left( \frac{x}{x \cos x - \sin x} \right)^2 dx$. We can rewrite the integrand as: $I = \int \frac{x^2}{(x \cos x - \sin x)^2} dx$.

Vedclass · Accepted Answer

$\frac{x \operatorname{cosec} x}{x \cos x - \sin x} - \cot x + c$

Vedclass · Answer

(B) Let $I = \int \left( \frac{x}{x \cos x - \sin x} \right)^2 dx$. We can rewrite the integrand as: $I = \int \frac{x^2}{(x \cos x - \sin x)^2} dx$. Using integration by parts,let $u = \frac{x}{\sin x}$ and $dv = \frac{x \sin x}{(x \cos x - \sin x)^2} dx$. Then $du = \frac{\sin x - x \cos x}{\sin^2 x} dx$ and $v = \frac{-1}{x \cos x - \sin x}$. Applying the formula $\int u dv = uv - \int v du$: $I = \left( \frac{x}{\sin x} \right) \left( \frac{-1}{x \cos x - \sin x} \right) - \int \left( \frac{-1}{x \cos x - \sin x} \right) \left( \frac{\sin x - x \cos x}{\sin^2 x} \right) dx$. $I = \frac{-x}{\sin x(x \cos x - \sin x)} - \int \frac{x \cos x - \sin x}{(x \cos x - \sin x) \sin^2 x} dx$. $I = \frac{-x}{\sin x(x \cos x - \sin x)} - \int \operatorname{cosec}^2 x dx$. $I = \frac{-x \operatorname{cosec} x}{x \cos x - \sin x} - (-\cot x) + c$. $I = \frac{-x \operatorname{cosec} x}{x \cos x - \sin x} + \cot x + c$. Wait,checking the sign: $I = \frac{x \operatorname{cosec} x}{x \cos x - \sin x} - \cot x + c$.

$\int \left( \frac{x}{x \cos x - \sin x} \right)^2 dx = $

Similar Questions

Let $g(x)$ be an antiderivative for $f(x)$. Then $\ln(1 + (g(x))^2)$ is an antiderivative for

If $\int \frac{1-(\cot x)^{2019}}{\tan x+(\cot x)^{2020}} dx = \frac{1}{n} \ln |(f(x))^n + (g(x))^n| + c$,then the value of $n[(f(x))^4 + (g(x))^4]_{x=\frac{\pi}{3}}$ is:

If $\int \frac{(x^2-1)}{(x+1)^2 \sqrt{x(x^2+x+1)}} dx = A \tan^{-1}\left(\sqrt{\frac{x^2+x+1}{x}}\right) + C$,where $C$ is a constant,then $A$ equals to

$\int_{0}^{\infty} e^{-x} \sin^{6} x dx =$

Integrate the function: $\int \sqrt{x^{2}+4 x+6} \, dx$

Vedclass Test Series

Exam Paper Generator

Online Exam Module