What is the integral of sin^6(v)cos^3(v) ?

The integral of sin^6(v)cos^3(v) is (sin^7(v))/7-(sin^9(v))/9+C

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integral of sin^6(v)cos^3(v)

\int sin^{6} (v) cos^{3} (v) d v

\frac{sin ^{7} ( v )}{7} - \frac{sin ^{9} ( v )}{9} + C

Solution steps

\int sin^{6} (v) cos^{3} (v) d v

Rewrite using trig identities

= \int (1 - sin^{2} (v)) cos (v) sin^{6} (v) d v

Apply u-substitution

: \int u^{6} (1 - u^{2}) d u

= \int u^{6} (1 - u^{2}) d u

Expand

u^{6} (1 - u^{2}) : u^{6} - u^{8}

= \int u^{6} - u^{8} 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^{6} d u - \int u^{8} d u

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

\int u^{8} d u = \frac{u ^{9}}{9}

= \frac{u ^{7}}{7} - \frac{u ^{9}}{9}

Substitute back

u = sin (v)

= \frac{sin ^{7} ( v )}{7} - \frac{sin ^{9} ( v )}{9}

Add a constant to the solution

= \frac{sin ^{7} ( v )}{7} - \frac{sin ^{9} ( v )}{9} + C

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What is the integral of sin^6(v)cos^3(v) ?
The integral of sin^6(v)cos^3(v) is (sin^7(v))/7-(sin^9(v))/9+C