What is the integral from pi/2 to pi of (1+cos(x))^2 ?

The integral from pi/2 to pi of (1+cos(x))^2 is pi-pi/4-2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from pi/2 to pi of (1+cos(x))^2

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

π - \frac{π}{4} - 2

Decimal

0.35619 \dots

Solution steps

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

Expand

(1 + cos (x))^{2} : 1 + 2 cos (x) + cos^{2} (x)

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

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

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

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

\int_{\frac{π}{2}}^{π} cos^{2} (x) d x = \frac{π}{4}

= π - \frac{π}{2} - 2 + \frac{π}{4}

Simplify

π - \frac{π}{2} - 2 + \frac{π}{4} : π - \frac{π}{4} - 2

= π - \frac{π}{4} - 2

What is the integral from pi/2 to pi of (1+cos(x))^2 ?
The integral from pi/2 to pi of (1+cos(x))^2 is pi-pi/4-2