What is the integral of-15x^4(-3x^5+6)^4 ?

The integral of-15x^4(-3x^5+6)^4 is -1/5 (3x^5-6)^5+C

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integral of-15x^4(-3x^5+6)^4

\int - 15 x^{4} (- 3 x^{5} + 6)^{4} d x

- \frac{1}{5} (3 x^{5} - 6)^{5} + C

Solution steps

\int - 15 x^{4} (- 3 x^{5} + 6)^{4} d x

Take the constant out:

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

= - 15 \cdot \int x^{4} (- 3 x^{5} + 6)^{4} d x

Factor

- 3 x^{5} + 6 : - (3 x^{5} - 6)

= - 15 \cdot \int x^{4} (- (3 x^{5} - 6))^{4} d x

(- (3 x^{5} - 6))^{4} = (3 x^{5} - 6)^{4}

= - 15 \cdot \int x^{4} (3 x^{5} - 6)^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{15} d u

= - 15 \cdot \int \frac{u ^{4}}{15} d u

Take the constant out:

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

= - 15 \cdot \frac{1}{15} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= - 15 \cdot \frac{1}{15} \cdot \frac{u ^{5}}{5}

Substitute back

u = 3 x^{5} - 6

= - 15 \cdot \frac{1}{15} \cdot \frac{( 3 x ^{5} - 6 ) ^{5}}{5}

Simplify

- 15 \cdot \frac{1}{15} \cdot \frac{( 3 x ^{5} - 6 ) ^{5}}{5} : - \frac{1}{5} (3 x^{5} - 6)^{5}

= - \frac{1}{5} (3 x^{5} - 6)^{5}

Add a constant to the solution

= - \frac{1}{5} (3 x^{5} - 6)^{5} + C

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What is the integral of-15x^4(-3x^5+6)^4 ?
The integral of-15x^4(-3x^5+6)^4 is -1/5 (3x^5-6)^5+C