int_0^pi sqrt{1+4 sin^2 frac{x}{2}+4 sin frac | vedclass

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

Question

(C) Let $I = \int_0^\pi \sqrt{1+4 \sin^2 \frac{x}{2} + 4 \sin \frac{x}{2}} \, dx$. Since $(1 + 2 \sin \frac{x}{2})^2 = 1 + 4 \sin^2 \frac{x}{2} + 4 \s

Vedclass · Accepted Answer

$\pi+4$

Vedclass · Answer

(C) Let $I = \int_0^\pi \sqrt{1+4 \sin^2 \frac{x}{2} + 4 \sin \frac{x}{2}} \, dx$. Since $(1 + 2 \sin \frac{x}{2})^2 = 1 + 4 \sin^2 \frac{x}{2} + 4 \sin \frac{x}{2}$, the integral becomes: $I = \int_0^\pi (1 + 2 \sin \frac{x}{2}) \, dx$. Integrating term by term: $I = [x - 2 \cdot 2 \cos \frac{x}{2}]_0^\pi$. $I = [x - 4 \cos \frac{x}{2}]_0^\pi$. Evaluating at the limits: $I = (\pi - 4 \cos \frac{\pi}{2}) - (0 - 4 \cos 0)$. $I = (\pi - 4(0)) - (0 - 4(1))$. $I = \pi - 0 + 4 = \pi + 4$.

$\int_0^\pi \sqrt{1+4 \sin^2 \frac{x}{2}+4 \sin \frac{x}{2}} \, dx$ is equal to

Similar Questions

If $\cos x + \cos 2x + \ldots + \cos nx = \frac{A(x)}{2 \sin(x/2)}$, then $\int_0^\pi A(x) dx =$

If $\int_{1}^{k}(3x^{2}+2x+1)dx=11$,then $k=$

The value of the definite integral $\int_{0}^{\frac{\pi}{2}} \sin x \sin 2x \sin 3x \, dx$ is equal to:

Let $0 < \alpha < \beta < 1$. Then $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{dx}{1+x}$ is

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module