What is the integral from 0 to pi/(12) of sin^2(3x) ?

The integral from 0 to pi/(12) of sin^2(3x) is (pi-2)/(24)

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/(12) of sin^2(3x)

\int_{0}^{\frac{π}{12}} sin^{2} (3 x) d x

\frac{π - 2}{24}

Decimal Notation

0.04756 \dots

Solution steps

\int_{0}^{\frac{π}{12}} sin^{2} (3 x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{12}} \frac{1}{2} (1 - cos (6 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}{12}} 1 - cos (6 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}{12}} 1 d x - \int_{0}^{\frac{π}{12}} cos (6 x) d x)

\int_{0}^{\frac{π}{12}} 1 d x = \frac{π}{12}

\int_{0}^{\frac{π}{12}} cos (6 x) d x = \frac{1}{6}

= \frac{1}{2} (\frac{π}{12} - \frac{1}{6})

Simplify

\frac{1}{2} (\frac{π}{12} - \frac{1}{6}) : \frac{π - 2}{24}

= \frac{π - 2}{24}

\int_{1}^{2} 5 x^{3} - 2 x d x

\int_{- 3}^{2} x + 1 d x

\int_{0}^{a} a x - x^{2} d x

\int_{- 3}^{0} 18 x + 55 d x

\int_{- 5}^{0} sin (\frac{nπ x}{5}) d x

What is the integral from 0 to pi/(12) of sin^2(3x) ?
The integral from 0 to pi/(12) of sin^2(3x) is (pi-2)/(24)