What is the integral of (sin(x))^3(cos(x))^{15} ?

The integral of (sin(x))^3(cos(x))^{15} is -(cos^{16}(x))/(16)+(cos^{18}(x))/(18)+C

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integral of (sin(x))^3(cos(x))^{15}

\int (sin (x))^{3} (cos (x))^{15} d x

- \frac{cos ^{16} ( x )}{16} + \frac{cos ^{18} ( x )}{18} + C

Solution steps

\int (sin (x))^{3} (cos (x))^{15} d x

Rewrite using trig identities

= \int (1 - cos^{2} (x)) sin (x) cos^{15} (x) d x

Apply u-substitution

: \int - u^{15} (1 - u^{2}) d u

= \int - u^{15} (1 - u^{2}) d u

Expand

- u^{15} (1 - u^{2}) : - u^{15} + u^{17}

= \int - u^{15} + u^{17} 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^{15} d u + \int u^{17} d u

\int u^{15} d u = \frac{u ^{16}}{16}

\int u^{17} d u = \frac{u ^{18}}{18}

= - \frac{u ^{16}}{16} + \frac{u ^{18}}{18}

Substitute back

u = cos (x)

= - \frac{cos ^{16} ( x )}{16} + \frac{cos ^{18} ( x )}{18}

Add a constant to the solution

= - \frac{cos ^{16} ( x )}{16} + \frac{cos ^{18} ( x )}{18} + C

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What is the integral of (sin(x))^3(cos(x))^{15} ?
The integral of (sin(x))^3(cos(x))^{15} is -(cos^{16}(x))/(16)+(cos^{18}(x))/(18)+C