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

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

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 3sin^3(x)

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

2

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

: \int_{1}^{0} - 1 + u^{2} d u

= 3 \cdot \int_{1}^{0} - 1 + u^{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 3 (- \int_{0}^{1} - 1 + u^{2} d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (- (- \int_{0}^{1} 1 d u + \int_{0}^{1} u^{2} d u))

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} u^{2} d u = \frac{1}{3}

= 3 (- (- 1 + \frac{1}{3}))

Simplify

3 (- (- 1 + \frac{1}{3})) : 2

= 2

\int_{1}^{4} \frac{ln ( x )}{x ^{6}} d x

\int_{1}^{16} x^{\frac{3}{2}} d x

\int_{0}^{2} [x^{2}] d x

\int_{1}^{2} \frac{1}{( 5 - 2 x ) ^{2}} d x

\int_{0}^{4} (30 - x) d x

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