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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

\int_{0}^{π} (x - sin^{2} (x)) d x

\frac{π ^{2} - π}{2}

Decimal

3.36400 \dots

Solution steps

\int_{0}^{π} x - sin^{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

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

\int_{0}^{π} x d x = \frac{π ^{2}}{2}

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

= \frac{π ^{2}}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π ^{2} - π}{2}

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