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

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

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integral of cos(2θ)(sin(θ)+cos(θ))^2

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

\frac{1}{4} (sin (θ) + cos (θ))^{4} + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{2} d u

= \int \frac{u}{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 u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = (sin (θ) + cos (θ))^{2}

= \frac{1}{2} \cdot \frac{( ( sin ( θ ) + cos ( θ ) ) ^{2} ) ^{2}}{2}

Simplify

\frac{1}{2} \cdot \frac{( ( sin ( θ ) + cos ( θ ) ) ^{2} ) ^{2}}{2} : \frac{1}{4} (sin (θ) + cos (θ))^{4}

= \frac{1}{4} (sin (θ) + cos (θ))^{4}

Add a constant to the solution

= \frac{1}{4} (sin (θ) + cos (θ))^{4} + C

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