What is the integral of 2x^3\sqrt[3]{x^4+1} ?

The integral of 2x^3\sqrt[3]{x^4+1} is 3/8 (x^4+1)^{4/3}+C

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integral of 2x^3\sqrt[3]{x^4+1}

\int 2 x^{3} 3 x^{4} + 1 d x

\frac{3}{8} (x^{4} + 1)^{\frac{4}{3}} + C

Solution steps

\int 2 x^{3} 3 x^{4} + 1 d x

Take the constant out:

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

= 2 \cdot \int x^{3} 3 x^{4} + 1 d x

Apply u-substitution

: \int \frac{3 u}{4} d u

= 2 \cdot \int \frac{3 u}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int 3 u d u

Apply the Power Rule

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

= 2 \cdot \frac{1}{4} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = x^{4} + 1

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

Simplify

2 \cdot \frac{1}{4} \cdot \frac{3}{4} (x^{4} + 1)^{\frac{4}{3}} : \frac{3}{8} (x^{4} + 1)^{\frac{4}{3}}

= \frac{3}{8} (x^{4} + 1)^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{8} (x^{4} + 1)^{\frac{4}{3}} + C

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What is the integral of 2x^3\sqrt[3]{x^4+1} ?
The integral of 2x^3\sqrt[3]{x^4+1} is 3/8 (x^4+1)^{4/3}+C