What is the integral of 5x^2(2x^3+6)^8 ?

The integral of 5x^2(2x^3+6)^8 is 5/54 (2x^3+6)^9+C

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integral of 5x^2(2x^3+6)^8

\int 5 x^{2} (2 x^{3} + 6)^{8} d x

\frac{5}{54} (2 x^{3} + 6)^{9} + C

Solution steps

\int 5 x^{2} (2 x^{3} + 6)^{8} d x

Take the constant out:

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

= 5 \cdot \int x^{2} (2 x^{3} + 6)^{8} d x

Apply u-substitution

: \int \frac{u ^{8}}{6} d u

= 5 \cdot \int \frac{u ^{8}}{6} d u

Take the constant out:

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

= 5 \cdot \frac{1}{6} \cdot \int u^{8} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{6} \cdot \frac{u ^{9}}{9}

Substitute back

u = 2 x^{3} + 6

= 5 \cdot \frac{1}{6} \cdot \frac{( 2 x ^{3} + 6 ) ^{9}}{9}

Simplify

5 \cdot \frac{1}{6} \cdot \frac{( 2 x ^{3} + 6 ) ^{9}}{9} : \frac{5}{54} (2 x^{3} + 6)^{9}

= \frac{5}{54} (2 x^{3} + 6)^{9}

Add a constant to the solution

= \frac{5}{54} (2 x^{3} + 6)^{9} + C

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What is the integral of 5x^2(2x^3+6)^8 ?
The integral of 5x^2(2x^3+6)^8 is 5/54 (2x^3+6)^9+C