What is the integral of 2x^2(x^3+4)^2 ?

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

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integral of 2x^2(x^3+4)^2

\int 2 x^{2} (x^{3} + 4)^{2} d x

\frac{2}{9} (x^{3} + 4)^{3} + C

Solution steps

\int 2 x^{2} (x^{3} + 4)^{2} d x

Take the constant out:

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

= 2 \cdot \int x^{2} (x^{3} + 4)^{2} d x

Apply u-substitution

: \int \frac{u ^{2}}{3} d u

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

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int u^{2} d u

Apply the Power Rule

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

= 2 \cdot \frac{1}{3} \cdot \frac{u ^{3}}{3}

Substitute back

u = x^{3} + 4

= 2 \cdot \frac{1}{3} \cdot \frac{( x ^{3} + 4 ) ^{3}}{3}

Simplify

2 \cdot \frac{1}{3} \cdot \frac{( x ^{3} + 4 ) ^{3}}{3} : \frac{2}{9} (x^{3} + 4)^{3}

= \frac{2}{9} (x^{3} + 4)^{3}

Add a constant to the solution

= \frac{2}{9} (x^{3} + 4)^{3} + C

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