What is the integral of x^2(5x^3-7)^8 ?

The integral of x^2(5x^3-7)^8 is -1/135 (-5x^3+7)^9+C

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integral of x^2(5x^3-7)^8

\int x^{2} (5 x^{3} - 7)^{8} d x

- \frac{1}{135} (- 5 x^{3} + 7)^{9} + C

Solution steps

\int x^{2} (5 x^{3} - 7)^{8} d x

Factor

5 x^{3} - 7 : - (- 5 x^{3} + 7)

= \int x^{2} (- (- 5 x^{3} + 7))^{8} d x

(- (- 5 x^{3} + 7))^{8} = (- 5 x^{3} + 7)^{8}

= \int x^{2} (- 5 x^{3} + 7)^{8} d x

Apply u-substitution

: \int - \frac{u ^{8}}{15} d u

= \int - \frac{u ^{8}}{15} d u

Take the constant out:

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

= - \frac{1}{15} \cdot \int u^{8} d u

Apply the Power Rule

: \frac{u ^{9}}{9}

= - \frac{1}{15} \cdot \frac{u ^{9}}{9}

Substitute back

u = - 5 x^{3} + 7

= - \frac{1}{15} \cdot \frac{( - 5 x ^{3} + 7 ) ^{9}}{9}

Simplify

- \frac{1}{15} \cdot \frac{( - 5 x ^{3} + 7 ) ^{9}}{9} : - \frac{1}{135} (- 5 x^{3} + 7)^{9}

= - \frac{1}{135} (- 5 x^{3} + 7)^{9}

Add a constant to the solution

= - \frac{1}{135} (- 5 x^{3} + 7)^{9} + C

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What is the integral of x^2(5x^3-7)^8 ?
The integral of x^2(5x^3-7)^8 is -1/135 (-5x^3+7)^9+C