What is the integral of x^2(1+2x^3)^3 ?

The integral of x^2(1+2x^3)^3 is 1/24 (1+2x^3)^4+C

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integral of x^2(1+2x^3)^3

\int x^{2} (1 + 2 x^{3})^{3} d x

\frac{1}{24} (1 + 2 x^{3})^{4} + C

Solution steps

\int x^{2} (1 + 2 x^{3})^{3} d x

Apply u-substitution

= \int \frac{u ^{3}}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u^{3} d u

Apply the Power Rule

= \frac{1}{6} \cdot \frac{u ^{4}}{4}

Substitute back

u = 1 + 2 x^{3}

= \frac{1}{6} \cdot \frac{( 1 + 2 x ^{3} ) ^{4}}{4}

Simplify

\frac{1}{6} \cdot \frac{( 1 + 2 x ^{3} ) ^{4}}{4} : \frac{1}{24} (1 + 2 x^{3})^{4}

= \frac{1}{24} (1 + 2 x^{3})^{4}

Add a constant to the solution

= \frac{1}{24} (1 + 2 x^{3})^{4} + C

What is the integral of x^2(1+2x^3)^3 ?
The integral of x^2(1+2x^3)^3 is 1/24 (1+2x^3)^4+C