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

The integral of (3x^2+2x)(5x^3+5x^2+2)^8 is 1/45 (5x^3+5x^2+2)^9+C

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

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

\frac{1}{45} (5 x^{3} + 5 x^{2} + 2)^{9} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{8}}{5} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 5 x^{3} + 5 x^{2} + 2

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

Simplify

\frac{1}{5} \cdot \frac{( 5 x ^{3} + 5 x ^{2} + 2 ) ^{9}}{9} : \frac{1}{45} (5 x^{3} + 5 x^{2} + 2)^{9}

= \frac{1}{45} (5 x^{3} + 5 x^{2} + 2)^{9}

Add a constant to the solution

= \frac{1}{45} (5 x^{3} + 5 x^{2} + 2)^{9} + C

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What is the integral of (3x^2+2x)(5x^3+5x^2+2)^8 ?
The integral of (3x^2+2x)(5x^3+5x^2+2)^8 is 1/45 (5x^3+5x^2+2)^9+C