int_0^{2 pi} theta sin ^6 theta cos theta , d | vedclass

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

Question

(D) ધારો કે $I = \int_0^{2 \pi} 	heta \sin ^6 	heta \cos 	heta \, d	heta \quad \dots (1)$ ગુણધર્મ $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$ નો

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) ધારો કે $I = \int_0^{2 \pi} 	heta \sin ^6 	heta \cos 	heta \, d	heta \quad \dots (1)$ ગુણધર્મ $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$ નો ઉપયોગ કરતા: $I = \int_0^{2 \pi} (2 \pi - 	heta) \sin ^6 (2 \pi - 	heta) \cos (2 \pi - 	heta) \, d	heta$ કારણ કે $\sin(2 \pi - 	heta) = -\sin 	heta$ અને $\cos(2 \pi - 	heta) = \cos 	heta$,તેથી: $I = \int_0^{2 \pi} (2 \pi - 	heta) \sin ^6 	heta \cos 	heta \, d	heta$ $I = 2 \pi \int_0^{2 \pi} \sin ^6 	heta \cos 	heta \, d	heta - \int_0^{2 \pi} 	heta \sin ^6 	heta \cos 	heta \, d	heta$ $I = 2 \pi \int_0^{2 \pi} \sin ^6 	heta \cos 	heta \, d	heta - I$ $2I = 2 \pi \int_0^{2 \pi} \sin ^6 	heta \cos 	heta \, d	heta$ $I = \pi \int_0^{2 \pi} \sin ^6 	heta \cos 	heta \, d	heta$ ધારો કે $f(	heta) = \sin ^6 	heta \cos 	heta$. તો $f(2 \pi - 	heta) = \sin ^6 (2 \pi - 	heta) \cos (2 \pi - 	heta) = (-\sin 	heta)^6 \cos 	heta = \sin ^6 	heta \cos 	heta = f(	heta)$. ગુણધર્મ $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$ જો $f(2a-x) = f(x)$ હોય તો: $I = \pi \cdot 2 \int_0^{\pi} \sin ^6 	heta \cos 	heta \, d	heta = 2 \pi \int_0^{\pi} \sin ^6 	heta \cos 	heta \, d	heta$ ધારો કે $u = \sin 	heta$,તો $du = \cos 	heta \, d	heta$. જ્યારે $	heta = 0, u = 0$. જ્યારે $	heta = \pi, u = 0$. $I = 2 \pi \int_0^0 u^6 \, du = 0$.

$\int_0^{2 \pi} \theta \sin ^6 \theta \cos \theta \, d\theta$ ની કિંમત શોધો.

Similar Questions

$\sum_{r=1}^{20} \left( \sqrt{\pi \left( \int_0^r x |\sin \pi x| dx \right)} \right)$ નું મૂલ્ય . . . . . . છે.

$\int_0^{\infty} (x^{12} + x^{-12}) \frac{\log x}{x} dx =$

$\int_0^\pi x \log(\sin x) \, dx = $

$\int_0^\pi \frac{x \tan x}{\sec x+\cos x} \,d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module