What is the integral of cos^5(θ)sin(2θ) ?

The integral of cos^5(θ)sin(2θ) is -2/7 cos^7(θ)+C

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integral of cos^5(θ)sin(2θ)

\int cos^{5} (θ) sin (2 θ) d θ

- \frac{2}{7} cos^{7} (θ) + C

Solution steps

\int cos^{5} (θ) sin (2 θ) d θ

Rewrite using trig identities

= \int 2 cos^{6} (θ) sin (θ) d θ

Take the constant out:

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

= 2 \cdot \int cos^{6} (θ) sin (θ) d θ

Apply u-substitution

: \int - u^{6} d u

= 2 \cdot \int - u^{6} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{7}}{7}

= 2 (- \frac{u ^{7}}{7})

Substitute back

u = cos (θ)

= 2 (- \frac{cos ^{7} ( θ )}{7})

Simplify

2 (- \frac{cos ^{7} ( θ )}{7}) : - \frac{2}{7} cos^{7} (θ)

= - \frac{2}{7} cos^{7} (θ)

Add a constant to the solution

= - \frac{2}{7} cos^{7} (θ) + C

\int \frac{x ^{2}}{1 - x ^{2}} d x

\int 5 tan^{3} (x) d x

x \to 0 lim (\frac{2}{( x - 5 ) ^{6}})

- 3 y^{^{'}} + \frac{2 y}{x} = \frac{y ^{4}}{x ^{4}}

y^{^{''}} - 3 y^{^{'}} - 4 y = 12

What is the integral of cos^5(θ)sin(2θ) ?
The integral of cos^5(θ)sin(2θ) is -2/7 cos^7(θ)+C