What is the integral of x^2\sqrt[3]{7x^3+1} ?

The integral of x^2\sqrt[3]{7x^3+1} is 1/28 (7x^3+1)^{4/3}+C

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integral of x^2\sqrt[3]{7x^3+1}

\int x^{2} 3 7 x^{3} + 1 d x

\frac{1}{28} (7 x^{3} + 1)^{\frac{4}{3}} + C

Solution steps

\int x^{2} 3 7 x^{3} + 1 d x

Apply u-substitution

: \int \frac{3 u}{21} d u

= \int \frac{3 u}{21} d u

Take the constant out:

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

= \frac{1}{21} \cdot \int 3 u d u

Apply the Power Rule

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

= \frac{1}{21} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = 7 x^{3} + 1

= \frac{1}{21} \cdot \frac{3}{4} (7 x^{3} + 1)^{\frac{4}{3}}

Simplify

\frac{1}{21} \cdot \frac{3}{4} (7 x^{3} + 1)^{\frac{4}{3}} : \frac{1}{28} (7 x^{3} + 1)^{\frac{4}{3}}

= \frac{1}{28} (7 x^{3} + 1)^{\frac{4}{3}}

Add a constant to the solution

= \frac{1}{28} (7 x^{3} + 1)^{\frac{4}{3}} + C

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What is the integral of x^2\sqrt[3]{7x^3+1} ?
The integral of x^2\sqrt[3]{7x^3+1} is 1/28 (7x^3+1)^{4/3}+C