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

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

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integral of (x^3+1)^{1/3}x^2

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = x^{3} + 1

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

Simplify

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

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

Add a constant to the solution

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

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