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

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

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 4sin^2(2x)

\int_{0}^{π} 4 sin^{2} (2 x) d x

2 π

Decimal

6.28318 \dots

Solution steps

\int_{0}^{π} 4 sin^{2} (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{0}^{π} sin^{2} (2 x) d x

Rewrite using trig identities

= 4 \cdot \int_{0}^{π} \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

= 4 \cdot \frac{1}{2} \cdot \int_{0}^{π} 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

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

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (4 x) d x = 0

= 4 \cdot \frac{1}{2} (π - 0)

Simplify

4 \cdot \frac{1}{2} (π - 0) : 2 π

= 2 π

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