What is the integral of cos^5(t)sin^2(t) ?

The integral of cos^5(t)sin^2(t) is (sin^3(t))/3-(2sin^5(t))/5+(sin^7(t))/7+C

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integral of cos^5(t)sin^2(t)

\int cos^{5} (t) sin^{2} (t) d t

\frac{sin ^{3} ( t )}{3} - \frac{2 sin ^{5} ( t )}{5} + \frac{sin ^{7} ( t )}{7} + C

Solution steps

\int cos^{5} (t) sin^{2} (t) d t

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

u^{2} (1 - u^{2})^{2} : u^{2} - 2 u^{4} + u^{6}

= \int u^{2} - 2 u^{4} + u^{6} 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 2 u^{4} d u + \int u^{6} d u

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

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{u ^{3}}{3} - \frac{2 u ^{5}}{5} + \frac{u ^{7}}{7}

Substitute back

u = sin (t)

= \frac{sin ^{3} ( t )}{3} - \frac{2 sin ^{5} ( t )}{5} + \frac{sin ^{7} ( t )}{7}

Add a constant to the solution

= \frac{sin ^{3} ( t )}{3} - \frac{2 sin ^{5} ( t )}{5} + \frac{sin ^{7} ( t )}{7} + C

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What is the integral of cos^5(t)sin^2(t) ?
The integral of cos^5(t)sin^2(t) is (sin^3(t))/3-(2sin^5(t))/5+(sin^7(t))/7+C