int_0^{2 pi} sin ^6 x cos ^5 x , dx is equal | vedclass

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

Question

(C) Let $I = \int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$. Using the property $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ if $f(2a-x) = f(x)$,we che

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Let $I = \int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$. Using the property $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ if $f(2a-x) = f(x)$,we check $f(x) = \sin^6 x \cos^5 x$. Since $f(2\pi - x) = \sin^6(2\pi - x) \cos^5(2\pi - x) = (-\sin x)^6 (\cos x)^5 = \sin^6 x \cos^5 x = f(x)$,we have $I = 2 \int_0^{\pi} \sin^6 x \cos^5 x \, dx$. Now,using the property $\int_0^a f(x) \, dx = 0$ if $f(a-x) = -f(x)$,we check $f(\pi - x)$. $f(\pi - x) = \sin^6(\pi - x) \cos^5(\pi - x) = (\sin x)^6 (-\cos x)^5 = -\sin^6 x \cos^5 x = -f(x)$. Therefore,$\int_0^{\pi} \sin^6 x \cos^5 x \, dx = 0$. Hence,$I = 2 	imes 0 = 0$.

$\int_0^{2 \pi} \sin ^6 x \cos ^5 x \, dx$ is equal to

Similar Questions

If $[t]$ denotes the greatest integer $\leq t$,then the value of $\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+[x^3]} dx$ is:

$\int_0^{2\pi } {\frac{{\sin 2\theta }}{{a - b\cos \theta }}\,d\theta = } $

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

$\int_{1}^{3} \frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}} dx$ is equal to

The value of the integral $\int_{-\pi / 2}^{\pi / 2}\left(x^2+\ln \frac{\pi+x}{\pi-x}\right) \cos x \, dx$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module