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

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

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

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

- \frac{cos ^{3} ( z )}{3} + \frac{cos ^{5} ( z )}{5} + C

Solution steps

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

Rewrite using trig identities

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

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 = cos (z)

= - \frac{cos ^{3} ( z )}{3} + \frac{cos ^{5} ( z )}{5}

Add a constant to the solution

= - \frac{cos ^{3} ( z )}{3} + \frac{cos ^{5} ( z )}{5} + C

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