If int frac{x-sin x}{1+cos x} dx = x tan left | vedclass

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

Question

(A) Let $I = \int \frac{x-\sin x}{1+\cos x} dx$. Using the identities $1+\cos x = 2\cos^2(\frac{x}{2})$ and $\sin x = 2\sin(\frac{x}{2})\cos(\frac{x}{

Vedclass · Accepted Answer

$-4$

Vedclass · Answer

(A) Let $I = \int \frac{x-\sin x}{1+\cos x} dx$. Using the identities $1+\cos x = 2\cos^2(\frac{x}{2})$ and $\sin x = 2\sin(\frac{x}{2})\cos(\frac{x}{2})$,we have: $I = \int \frac{x}{2\cos^2(\frac{x}{2})} dx - \int \frac{2\sin(\frac{x}{2})\cos(\frac{x}{2})}{2\cos^2(\frac{x}{2})} dx$ $I = \frac{1}{2} \int x \sec^2(\frac{x}{2}) dx - \int 	an(\frac{x}{2}) dx$. Applying integration by parts to the first term: $\int x \sec^2(\frac{x}{2}) dx = x(2	an(\frac{x}{2})) - \int 1 \cdot 2	an(\frac{x}{2}) dx = 2x	an(\frac{x}{2}) - 2 \int 	an(\frac{x}{2}) dx$. Substituting this back into $I$: $I = \frac{1}{2} [2x	an(\frac{x}{2}) - 2 \int 	an(\frac{x}{2}) dx] - \int 	an(\frac{x}{2}) dx$ $I = x	an(\frac{x}{2}) - \int 	an(\frac{x}{2}) dx - \int 	an(\frac{x}{2}) dx = x	an(\frac{x}{2}) - 2 \int 	an(\frac{x}{2}) dx$. Since $\int 	an(\frac{x}{2}) dx = 2 \log |\sec(\frac{x}{2})|$,we get: $I = x	an(\frac{x}{2}) - 2(2 \log |\sec(\frac{x}{2})|) + C = x	an(\frac{x}{2}) - 4 \log |\sec(\frac{x}{2})| + C$. Comparing this with the given expression $x 	an(\frac{x}{2}) + p \log |\sec(\frac{x}{2})| + C$,we find $p = -4$.

If $\int \frac{x-\sin x}{1+\cos x} dx = x \tan \left(\frac{x}{2}\right) + p \log \left|\sec \left(\frac{x}{2}\right)\right| + C$,then $p$ is equal to

Similar Questions

If $f(x) = \int \operatorname{cosec}^5 x \, dx$,then $f\left(\frac{\pi}{4}\right) = $

$\int \frac{dx}{\tan x+\cot x+\sec x+\operatorname{cosec} x} = $

If $\int \frac{3e^x - 5e^{-x}}{4e^x + 5e^{-x}} dx = px + q \cdot \log |4e^x + 5e^{-x}| + C$, then

$\int \frac{25 x^2+8}{\sqrt{25 x^2+9}} d x=$

Find the integral of the function $\sin ^{4} x$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module