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

The integral of x^2(5+2x^3)^8 is 1/54 (5+2x^3)^9+C

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

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

\frac{1}{54} (5 + 2 x^{3})^{9} + C

Solution steps

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

Apply u-substitution

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

= \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

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

Apply the Power Rule

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

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

Substitute back

u = 5 + 2 x^{3}

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

Simplify

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

= \frac{1}{54} (5 + 2 x^{3})^{9}

Add a constant to the solution

= \frac{1}{54} (5 + 2 x^{3})^{9} + C

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What is the integral of x^2(5+2x^3)^8 ?
The integral of x^2(5+2x^3)^8 is 1/54 (5+2x^3)^9+C