What is the integral of sin^3(2x)cos^7(x) ?

The integral of sin^3(2x)cos^7(x) is 8(-(cos^{11}(x))/(11)+(cos^{13}(x))/(13))+C

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integral of sin^3(2x)cos^7(x)

\int sin^{3} (2 x) cos^{7} (x) d x

8 (- \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13}) + C

Solution steps

\int sin^{3} (2 x) cos^{7} (x) d x

Rewrite using trig identities

= \int 8 cos^{10} (x) sin^{3} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int cos^{10} (x) sin^{3} (x) d x

Rewrite using trig identities

= 8 \cdot \int (1 - cos^{2} (x)) sin (x) cos^{10} (x) d x

Apply u-substitution

: \int - u^{10} (1 - u^{2}) d u

= 8 \cdot \int - u^{10} (1 - u^{2}) d u

Expand

- u^{10} (1 - u^{2}) : - u^{10} + u^{12}

= 8 \cdot \int - u^{10} + u^{12} 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

= 8 (- \int u^{10} d u + \int u^{12} d u)

\int u^{10} d u = \frac{u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= 8 (- \frac{u ^{11}}{11} + \frac{u ^{13}}{13})

Substitute back

u = cos (x)

= 8 (- \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13})

Add a constant to the solution

= 8 (- \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13}) + C

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What is the integral of sin^3(2x)cos^7(x) ?
The integral of sin^3(2x)cos^7(x) is 8(-(cos^{11}(x))/(11)+(cos^{13}(x))/(13))+C