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

The integral of cos^2(2θ)sin^3(2θ) is 1/2 (-(cos^3(2θ))/3+(cos^5(2θ))/5)+C

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integral of cos^2(2θ)sin^3(2θ)

\int cos^{2} (2 θ) sin^{3} (2 θ) d θ

\frac{1}{2} (- \frac{cos ^{3} ( 2 θ )}{3} + \frac{cos ^{5} ( 2 θ )}{5}) + C

Solution steps

\int cos^{2} (2 θ) sin^{3} (2 θ) d θ

Apply u-substitution

= \int cos^{2} (u) sin^{3} (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int cos^{2} (u) sin^{3} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int (1 - cos^{2} (u)) sin (u) cos^{2} (u) d u

Apply u-substitution

= \frac{1}{2} \cdot \int - v^{2} (1 - v^{2}) d v

Expand

- v^{2} (1 - v^{2}) : - v^{2} + v^{4}

= \frac{1}{2} \cdot \int - v^{2} + v^{4} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int v^{2} d v + \int v^{4} d v)

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

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{2} (- \frac{v ^{3}}{3} + \frac{v ^{5}}{5})

Substitute back

= \frac{1}{2} (- \frac{cos ^{3} ( 2 θ )}{3} + \frac{cos ^{5} ( 2 θ )}{5})

Add a constant to the solution

= \frac{1}{2} (- \frac{cos ^{3} ( 2 θ )}{3} + \frac{cos ^{5} ( 2 θ )}{5}) + C

What is the integral of cos^2(2θ)sin^3(2θ) ?
The integral of cos^2(2θ)sin^3(2θ) is 1/2 (-(cos^3(2θ))/3+(cos^5(2θ))/5)+C