What is the integral from-pi to pi of (sin(x))^2 ?

The integral from-pi to pi of (sin(x))^2 is pi

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from-pi to pi of (sin(x))^2

\int_{- π}^{π} (sin (x))^{2} d x

π

Decimal

3.14159 \dots

Solution steps

\int_{- π}^{π} (sin (x))^{2} d x

Simplify

= \int_{- π}^{π} sin^{2} (x) d x

Rewrite using trig identities

= \int_{- π}^{π} \frac{1 - cos ( 2 x )}{2} 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_{- π}^{π} 1 - cos (2 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

= \frac{1}{2} (\int_{- π}^{π} 1 d x - \int_{- π}^{π} cos (2 x) d x)

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos (2 x) d x = 0

= \frac{1}{2} (2 π - 0)

Simplify

\frac{1}{2} (2 π - 0) : π

= π

What is the integral from-pi to pi of (sin(x))^2 ?
The integral from-pi to pi of (sin(x))^2 is pi