What is the integral of sin^3(3θ)cos^4(3θ) ?

The integral of sin^3(3θ)cos^4(3θ) is 1/3 (-(cos^5(3θ))/5+(cos^7(3θ))/7)+C

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integral of sin^3(3θ)cos^4(3θ)

\int sin^{3} (3 θ) cos^{4} (3 θ) d θ

\frac{1}{3} (- \frac{cos ^{5} ( 3 θ )}{5} + \frac{cos ^{7} ( 3 θ )}{7}) + C

Solution steps

\int sin^{3} (3 θ) cos^{4} (3 θ) d θ

Apply u-substitution

= \int sin^{3} (u) cos^{4} (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sin^{3} (u) cos^{4} (u) d u

Rewrite using trig identities

= \frac{1}{3} \cdot \int (1 - cos^{2} (u)) sin (u) cos^{4} (u) d u

Apply u-substitution

= \frac{1}{3} \cdot \int - v^{4} (1 - v^{2}) d v

Expand

- v^{4} (1 - v^{2}) : - v^{4} + v^{6}

= \frac{1}{3} \cdot \int - v^{4} + v^{6} 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}{3} (- \int v^{4} d v + \int v^{6} d v)

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

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

= \frac{1}{3} (- \frac{v ^{5}}{5} + \frac{v ^{7}}{7})

Substitute back

= \frac{1}{3} (- \frac{cos ^{5} ( 3 θ )}{5} + \frac{cos ^{7} ( 3 θ )}{7})

Add a constant to the solution

= \frac{1}{3} (- \frac{cos ^{5} ( 3 θ )}{5} + \frac{cos ^{7} ( 3 θ )}{7}) + C

What is the integral of sin^3(3θ)cos^4(3θ) ?
The integral of sin^3(3θ)cos^4(3θ) is 1/3 (-(cos^5(3θ))/5+(cos^7(3θ))/7)+C