What is the integral from 0 to a of sin^2((npix)/a) ?

The integral from 0 to a of sin^2((npix)/a) is 1/2 (a-(asin(2pin))/(2pin))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to a of sin^2((npix)/a)

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{a} 1 - cos (\frac{2 πn x}{a}) 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

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

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

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

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

What is the integral from 0 to a of sin^2((npix)/a) ?
The integral from 0 to a of sin^2((npix)/a) is 1/2 (a-(asin(2pin))/(2pin))