help@askiitians.com

+91-735-322-1155

Integral Calculus> Integral [sin(nx)]^2/[sin(x)]^2, upper li...

question mark

Integral [sin(nx)]^2/[sin(x)]^2, upper limit : pi, lower limit : 0

Abhishek Ghatge , 11 Years ago

Grade 12

1 Answers

Jitender Singh

Ans:

$I_{n} = \int_{0}^{\pi}\frac{sin^{2}(nx)}{sin^{2}(x)}dx$

$I_{1} = \int_{0}^{\pi}\frac{sin^{2}(x)}{sin^{2}(x)}dx$

$I_{1} = \pi$

$I_{2} = \int_{0}^{\pi}\frac{sin^{2}(2x)}{sin^{2}(x)}dx$

$I_{2} = \int_{0}^{\pi}\frac{(2sin(x).cos(x))^{2}}{sin^{2}(x)}dx$

$I_{2} = \int_{0}^{\pi}4cos^{2}(x)dx$ $I_{2} = \int_{0}^{\pi}2(1+cos(2x))dx = (2x + sin(2x))_{0}^{\pi} = 2\pi$

$I_{3} = \int_{0}^{\pi}\frac{sin^{2}(3x)}{sin^{2}(x)}dx$

$I_{3} = (3x + 2sin(2x) + \frac{1}{2}sin (4x))_{0}^{\pi}$

$I_{3} = 3\pi$

$I_{4} = \int_{0}^{\pi}\frac{sin^{2}(4x)}{sin^{2}(x)}dx$

$I_{4} = (4x+3sin(2x) + sin(4x) + \frac{1}{3}sin(6x))_{0}^{\pi}$

$I_{4} =4\pi$

Similarly, when we do for the n, we get

$I_{n} =n\pi$

Thanks & Regards

Jitender Singh

IIT Delhi

askIITians Faculty

Last Activity: 11 Years ago

LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

Full Live Access

Study Material

Live Doubts Solving

Daily Class Assignments

The area of a square field is 640000cm².what is its area in hectares

integral calculus

1 Answer Available

Last Activity: 3 Years ago

Please solve this integration. I am facing difficulty in it
How to solve this. Please solve anyone. Humble request

integral calculus

1 Answer Available

Last Activity: 3 Years ago

Using laplace transform, solve d²y/dt²+ dy/dt = t² +2t given that y=4 and y’=-2 and t=0

integral calculus

1 Answer Available

Last Activity: 3 Years ago

If 4th term of an A.p is zero, show that the 8th term is double the 6th term

integral calculus

1 Answer Available

Last Activity: 3 Years ago

View all Questions