What is the integral from 0 to 1 of sin^2(ax) ?

The integral from 0 to 1 of sin^2(ax) is 1-1/2 (1+(sin(2a))/(2a))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of sin^2(ax)

\int_{0}^{1} sin^{2} (a x) d x

1 - \frac{1}{2} (1 + \frac{sin ( 2 a )}{2 a})

Solution steps

\int_{0}^{1} sin^{2} (a x) d x

Rewrite using trig identities

= \int_{0}^{1} 1 - cos^{2} (a x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} 1 d x - \int_{0}^{1} cos^{2} (a x) d x

\int_{0}^{1} 1 d x = 1

\int_{0}^{1} cos^{2} (a x) d x = \frac{1}{2} (1 + \frac{sin ( 2 a )}{2 a})

= 1 - \frac{1}{2} (1 + \frac{sin ( 2 a )}{2 a})

\int_{0}^{2} (\frac{1}{2} + y + z) d y

\int_{0}^{4} 2 (3)^{x} d x

\int_{1}^{2} x^{2} + \frac{1}{x ^{2}} d x

\int_{0}^{1} 4 - 2 x^{2} d x

\int_{0}^{2} (x - 3)^{3} d x

What is the integral from 0 to 1 of sin^2(ax) ?
The integral from 0 to 1 of sin^2(ax) is 1-1/2 (1+(sin(2a))/(2a))