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

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

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integral of cos(θ)sin(2θ)

\int cos (θ) sin (2 θ) d θ

- \frac{2}{3} cos^{3} (θ) + C

Solution steps

\int cos (θ) sin (2 θ) d θ

Rewrite using trig identities

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

Apply u-substitution

: \int - u^{2} d u

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{3}}{3}

= 2 (- \frac{u ^{3}}{3})

Substitute back

u = cos (θ)

= 2 (- \frac{cos ^{3} ( θ )}{3})

Simplify

2 (- \frac{cos ^{3} ( θ )}{3}) : - \frac{2}{3} cos^{3} (θ)

= - \frac{2}{3} cos^{3} (θ)

Add a constant to the solution

= - \frac{2}{3} cos^{3} (θ) + C

\int (b z^{2} + \frac{c}{z}) d z

\frac{d}{d x} ((sin (2 x))^{\frac{c o s ( x )}{x ^{3}}})

cos (3^{x})

\frac{d}{d x} (\frac{1 - 2 x}{x})

\frac{d}{d x} (cos (x) - e^{x})

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