What is the integral of x\sqrt[3]{81+x^2} ?

The integral of x\sqrt[3]{81+x^2} is 3/8 (81+x^2)^{4/3}+C

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integral of x\sqrt[3]{81+x^2}

\int x 3 81 + x^{2} d x

\frac{3}{8} (81 + x^{2})^{\frac{4}{3}} + C

Solution steps

\int x 3 81 + x^{2} d x

Apply u-substitution

: \int \frac{3 u}{2} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 81 + x^{2}

= \frac{1}{2} \cdot \frac{3}{4} (81 + x^{2})^{\frac{4}{3}}

Simplify

\frac{1}{2} \cdot \frac{3}{4} (81 + x^{2})^{\frac{4}{3}} : \frac{3}{8} (81 + x^{2})^{\frac{4}{3}}

= \frac{3}{8} (81 + x^{2})^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{8} (81 + x^{2})^{\frac{4}{3}} + C

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\int 20 e^{u} d u

\frac{d}{d x} (e^{x} - 4 x^{2})

x^{2} + x + 3

\int \frac{1}{2 y ^{2}} d y

What is the integral of x\sqrt[3]{81+x^2} ?
The integral of x\sqrt[3]{81+x^2} is 3/8 (81+x^2)^{4/3}+C