What is the integral of pisin(pix)(3+2cos(pix))^3 ?

The integral of pisin(pix)(3+2cos(pix))^3 is -1/8 (3+2cos(pix))^4+C

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integral of pisin(pix)(3+2cos(pix))^3

\int π sin (π x) (3 + 2 cos (π x))^{3} d x

- \frac{1}{8} (3 + 2 cos (π x))^{4} + C

Solution steps

\int π sin (π x) (3 + 2 cos (π x))^{3} d x

Take the constant out:

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

= π \cdot \int sin (π x) (3 + 2 cos (π x))^{3} d x

Apply u-substitution

: \int - \frac{u ^{3}}{2 π} d u

= π \cdot \int - \frac{u ^{3}}{2 π} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{4}}{4}

= π (- \frac{1}{2 π} \cdot \frac{u ^{4}}{4})

Substitute back

u = 3 + 2 cos (π x)

= π (- \frac{1}{2 π} \cdot \frac{( 3 + 2 cos ( π x ) ) ^{4}}{4})

Simplify

π (- \frac{1}{2 π} \cdot \frac{( 3 + 2 cos ( π x ) ) ^{4}}{4}) : - \frac{1}{8} (3 + 2 cos (π x))^{4}

= - \frac{1}{8} (3 + 2 cos (π x))^{4}

Add a constant to the solution

= - \frac{1}{8} (3 + 2 cos (π x))^{4} + C

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What is the integral of pisin(pix)(3+2cos(pix))^3 ?
The integral of pisin(pix)(3+2cos(pix))^3 is -1/8 (3+2cos(pix))^4+C