What is the integral of cos^3(θ)sin^2(θ) ?

The integral of cos^3(θ)sin^2(θ) is (sin^3(θ))/3-(sin^5(θ))/5+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cos^3(θ)sin^2(θ)

\int cos^{3} (θ) sin^{2} (θ) d θ

\frac{sin ^{3} ( θ )}{3} - \frac{sin ^{5} ( θ )}{5} + C

Solution steps

\int cos^{3} (θ) sin^{2} (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

= \int u^{2} (1 - u^{2}) d u

Expand

u^{2} (1 - u^{2}) : u^{2} - u^{4}

= \int u^{2} - u^{4} 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 u^{2} d u - \int u^{4} d u

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

\int u^{4} d u = \frac{u ^{5}}{5}

= \frac{u ^{3}}{3} - \frac{u ^{5}}{5}

Substitute back

u = sin (θ)

= \frac{sin ^{3} ( θ )}{3} - \frac{sin ^{5} ( θ )}{5}

Add a constant to the solution

= \frac{sin ^{3} ( θ )}{3} - \frac{sin ^{5} ( θ )}{5} + C

What is the integral of cos^3(θ)sin^2(θ) ?
The integral of cos^3(θ)sin^2(θ) is (sin^3(θ))/3-(sin^5(θ))/5+C