What is the integral from 0 to pi of (sin^2(x))/2 ?

The integral from 0 to pi of (sin^2(x))/2 is pi/4

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi of (sin^2(x))/2

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

\frac{π}{4}

Decimal

0.78539 \dots

Solution steps

\int_{0}^{π} \frac{sin ^{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_{0}^{π} sin^{2} (x) d x

Rewrite using trig identities

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

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (2 x) d x = 0

= \frac{1}{2} \cdot \frac{1}{2} (π - 0)

Simplify

\frac{1}{2} \cdot \frac{1}{2} (π - 0) : \frac{π}{4}

= \frac{π}{4}

\int_{1}^{5} π (x - 1)^{2} d x

\int_{1}^{e^{3}} \frac{1 - ln ( x )}{x} d x

\int_{0}^{1} e^{2 t} d t

\int_{1}^{2} 3^{x} d x

\int_{0}^{1} (x^{2} - 3 x + 7) d x

What is the integral from 0 to pi of (sin^2(x))/2 ?
The integral from 0 to pi of (sin^2(x))/2 is pi/4