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

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

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

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

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

Solution steps

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

Simplify

= \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

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

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