What is the integral from 0 to pi/3 of 81sin^2(x) ?

The integral from 0 to pi/3 of 81sin^2(x) is (27(4pi-3sqrt(3)))/8

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/3 of 81sin^2(x)

\int_{0}^{\frac{π}{3}} 81 sin^{2} (x) d x

\frac{27 ( 4 π - 3 3 )}{8}

Decimal Notation

24.87448 \dots

Solution steps

\int_{0}^{\frac{π}{3}} 81 sin^{2} (x) d x

Take the constant out:

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

= 81 \cdot \int_{0}^{\frac{π}{3}} sin^{2} (x) d x

Rewrite using trig identities

= 81 \cdot \int_{0}^{\frac{π}{3}} \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

= 81 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{3}} 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

= 81 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{3}} 1 d x - \int_{0}^{\frac{π}{3}} cos (2 x) d x)

\int_{0}^{\frac{π}{3}} 1 d x = \frac{π}{3}

\int_{0}^{\frac{π}{3}} cos (2 x) d x = \frac{3}{4}

= 81 \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4})

Simplify

81 \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4}) : \frac{27 ( 4 π - 3 3 )}{8}

= \frac{27 ( 4 π - 3 3 )}{8}

\int \frac{1}{( x - 2 ) ^{2} ( x ^{2} + 3 ) ^{2}} d x

\frac{d}{d x} (e^{- \frac{x}{2}} cos (\frac{3}{2} x))

\frac{d}{d x} (- \frac{3}{2} x^{4} cos (x))

x^{2}, x

\int_{0}^{5} x^{3} d x

What is the integral from 0 to pi/3 of 81sin^2(x) ?
The integral from 0 to pi/3 of 81sin^2(x) is (27(4pi-3sqrt(3)))/8