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

The integral of x^2(4x^3+2)^3 is 1/48 (4x^3+2)^4+C

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

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

\frac{1}{48} (4 x^{3} + 2)^{4} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{\frac{1}{3}}}{36} d u

= \int \frac{u ^{\frac{1}{3}}}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int u^{\frac{1}{3}} d u

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{36} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = (4 x^{3} + 2)^{3}

= \frac{1}{36} \cdot \frac{3}{4} ((4 x^{3} + 2)^{3})^{\frac{4}{3}}

Simplify

\frac{1}{36} \cdot \frac{3}{4} ((4 x^{3} + 2)^{3})^{\frac{4}{3}} : \frac{1}{48} (4 x^{3} + 2)^{4}

= \frac{1}{48} (4 x^{3} + 2)^{4}

Add a constant to the solution

= \frac{1}{48} (4 x^{3} + 2)^{4} + C

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What is the integral of x^2(4x^3+2)^3 ?
The integral of x^2(4x^3+2)^3 is 1/48 (4x^3+2)^4+C