What is the integral of y^2(y^3-1)^7 ?

The integral of y^2(y^3-1)^7 is 1/24 (y^3-1)^8+C

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integral of y^2(y^3-1)^7

\int y^{2} (y^{3} - 1)^{7} d y

\frac{1}{24} (y^{3} - 1)^{8} + C

Solution steps

\int y^{2} (y^{3} - 1)^{7} d y

Apply u-substitution

: \int \frac{u ( u ^{\frac{3}{2}} - 1 ) ^{7}}{2 y} d u

= \int \frac{( u - 1 ) ^{7}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int (u - 1)^{7} d u

Apply u-substitution

: \int v^{7} d v

= \frac{1}{3} \cdot \int v^{7} d v

Apply the Power Rule

: \frac{v ^{8}}{8}

= \frac{1}{3} \cdot \frac{v ^{8}}{8}

Substitute back

= \frac{1}{3} \cdot \frac{( y ^{3} - 1 ) ^{8}}{8}

Simplify

\frac{1}{3} \cdot \frac{( y ^{3} - 1 ) ^{8}}{8} : \frac{1}{24} (y^{3} - 1)^{8}

= \frac{1}{24} (y^{3} - 1)^{8}

Add a constant to the solution

= \frac{1}{24} (y^{3} - 1)^{8} + C

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What is the integral of y^2(y^3-1)^7 ?
The integral of y^2(y^3-1)^7 is 1/24 (y^3-1)^8+C