What is the integral of cos^6(θ)sin(2θ) ?

The integral of cos^6(θ)sin(2θ) is -1/4 cos^8(θ)+C

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integral of cos^6(θ)sin(2θ)

\int cos^{6} (θ) sin (2 θ) d θ

- \frac{1}{4} cos^{8} (θ) + C

Solution steps

\int cos^{6} (θ) sin (2 θ) d θ

Rewrite using trig identities

= \int 2 cos^{7} (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^{7} (θ) sin (θ) d θ

Apply u-substitution

: \int - u^{7} d u

= 2 \cdot \int - u^{7} d u

Take the constant out:

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

= 2 (- \int u^{7} d u)

Apply the Power Rule

: \frac{u ^{8}}{8}

= 2 (- \frac{u ^{8}}{8})

Substitute back

u = cos (θ)

= 2 (- \frac{cos ^{8} ( θ )}{8})

Simplify

2 (- \frac{cos ^{8} ( θ )}{8}) : - \frac{1}{4} cos^{8} (θ)

= - \frac{1}{4} cos^{8} (θ)

Add a constant to the solution

= - \frac{1}{4} cos^{8} (θ) + C

\int x 16 - 4 x^{2} d x

\frac{d}{d x} (5 ln (12 x))

x \to 0 lim (\frac{x ^{3} - 1}{x})

x \to 3 lim (\frac{2 x}{x - 3})

\frac{d y}{d x} = e^{x + y - 1} - 1

What is the integral of cos^6(θ)sin(2θ) ?
The integral of cos^6(θ)sin(2θ) is -1/4 cos^8(θ)+C