What is the integral of (3x^3+9x)^3(3x^2+3) ?

The integral of (3x^3+9x)^3(3x^2+3) is 1/12 (3x^3+9x)^4+C

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integral of (3x^3+9x)^3(3x^2+3)

\int (3 x^{3} + 9 x)^{3} (3 x^{2} + 3) d x

\frac{1}{12} (3 x^{3} + 9 x)^{4} + C

Solution steps

\int (3 x^{3} + 9 x)^{3} (3 x^{2} + 3) d x

Apply u-substitution

: \int \frac{u ^{3}}{3} d u

= \int \frac{u ^{3}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{1}{3} \cdot \frac{u ^{4}}{4}

Substitute back

u = 3 x^{3} + 9 x

= \frac{1}{3} \cdot \frac{( 3 x ^{3} + 9 x ) ^{4}}{4}

Simplify

\frac{1}{3} \cdot \frac{( 3 x ^{3} + 9 x ) ^{4}}{4} : \frac{1}{12} (3 x^{3} + 9 x)^{4}

= \frac{1}{12} (3 x^{3} + 9 x)^{4}

Add a constant to the solution

= \frac{1}{12} (3 x^{3} + 9 x)^{4} + C

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What is the integral of (3x^3+9x)^3(3x^2+3) ?
The integral of (3x^3+9x)^3(3x^2+3) is 1/12 (3x^3+9x)^4+C