What is the integral of sin^3(X) ?

The integral of sin^3(X) is -cos(X)+(cos^3(X))/3+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of sin^3(X)

\int sin^{3} (X) d X

- cos (X) + \frac{cos ^{3} ( X )}{3} + C

Solution steps

\int sin^{3} (X) d X

Rewrite using trig identities

= \int (1 - cos^{2} (X)) sin (X) d X

Apply u-substitution

= \int - 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

= - \int 1 d u + \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= - u + \frac{u ^{3}}{3}

Substitute back

u = cos (X)

= - cos (X) + \frac{cos ^{3} ( X )}{3}

Add a constant to the solution

= - cos (X) + \frac{cos ^{3} ( X )}{3} + C

What is the integral of sin^3(X) ?
The integral of sin^3(X) is -cos(X)+(cos^3(X))/3+C