What is the integral of (9x^2+12)(2x^3+8x)^{11} ?

The integral of (9x^2+12)(2x^3+8x)^{11} is 1/8 (2x^3+8x)^{12}+C

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integral of (9x^2+12)(2x^3+8x)^{11}

\int (9 x^{2} + 12) (2 x^{3} + 8 x)^{11} d x

\frac{1}{8} (2 x^{3} + 8 x)^{12} + C

Solution steps

\int (9 x^{2} + 12) (2 x^{3} + 8 x)^{11} d x

Apply u-substitution

= \int \frac{3 u ^{11}}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int u^{11} d u

Apply the Power Rule

= \frac{3}{2} \cdot \frac{u ^{12}}{12}

Substitute back

u = 2 x^{3} + 8 x

= \frac{3}{2} \cdot \frac{( 2 x ^{3} + 8 x ) ^{12}}{12}

Simplify

\frac{3}{2} \cdot \frac{( 2 x ^{3} + 8 x ) ^{12}}{12} : \frac{1}{8} (2 x^{3} + 8 x)^{12}

= \frac{1}{8} (2 x^{3} + 8 x)^{12}

Add a constant to the solution

= \frac{1}{8} (2 x^{3} + 8 x)^{12} + C

What is the integral of (9x^2+12)(2x^3+8x)^{11} ?
The integral of (9x^2+12)(2x^3+8x)^{11} is 1/8 (2x^3+8x)^{12}+C