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

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

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

\frac{π}{4}

Decimal Notation

0.78539 \dots

Solution steps

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

Rewrite using trig identities

= \int_{0}^{\frac{π}{2}} \frac{1}{2} (1 - cos (4 x)) 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}^{\frac{π}{2}} 1 - cos (4 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_{0}^{\frac{π}{2}} 1 d x - \int_{0}^{\frac{π}{2}} cos (4 x) d x)

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (4 x) d x = 0

= \frac{1}{2} (\frac{π}{2} - 0)

Simplify

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

= \frac{π}{4}

\int (3 x - 6) d x

x \to 5 - lim (\frac{2}{( x - 5 ) ^{3}})

\frac{d}{d x} (\frac{sin ( x )}{3 x ^{2}})

x \to 3 lim (g (x))

g (x) = (x - 4)^{3} + 3 x (x - 4)^{2}

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