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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{π}{2}

Decimal Notation

1.57079 \dots

Solution steps

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

Simplify

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

Rewrite using trig identities

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

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

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

= \frac{1}{2} (π + 0)

Simplify

\frac{1}{2} (π + 0) : \frac{π}{2}

= \frac{π}{2}

\int_{3}^{4} 1

\int_{0}^{9} t d t

\int_{- \frac{π}{4}}^{\frac{π}{4}} 2 sec^{2} (x) d x

\int_{1}^{e^{2}} \frac{1}{x ( 1 + ln ( x ) )} d x

\int_{0}^{6} 9 - (x - 3)^{2} d x

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