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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cos^3(x)sin^2(x)

\int cos^{3} (x) sin^{2} (x) d x

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

Solution steps

\int cos^{3} (x) sin^{2} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

= \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 (x)

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

Add a constant to the solution

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

\int x^{5} 15 + x^{6} d x

\frac{\partial}{\partial z} (x e^{- z})

\frac{\partial}{\partial z} (x sin (y - z))

x \to 0 lim (5 e)

\frac{d}{d x} (\frac{9}{x + 7})

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