What is the integral of (2x^3+5)^2(9x^2) ?

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

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integral of (2x^3+5)^2(9x^2)

\int (2 x^{3} + 5)^{2} (9 x^{2}) d x

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

Solution steps

\int (2 x^{3} + 5)^{2} \cdot 9 x^{2} d x

Take the constant out:

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

= 9 \cdot \int (2 x^{3} + 5)^{2} x^{2} d x

Apply u-substitution

: \int \frac{u ^{2}}{6} d u

= 9 \cdot \int \frac{u ^{2}}{6} d u

Take the constant out:

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

= 9 \cdot \frac{1}{6} \cdot \int u^{2} d u

Apply the Power Rule

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

= 9 \cdot \frac{1}{6} \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 x^{3} + 5

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

Simplify

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

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

Add a constant to the solution

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

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