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

The integral of x^2*sqrt((2x^3+1)) is 1/9 (2x^3+1)^{3/2}+C

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

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

\frac{1}{9} (2 x^{3} + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

= \int \frac{u}{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 d u

Apply the Power Rule

= \frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 2 x^{3} + 1

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

Simplify

\frac{1}{6} \cdot \frac{2}{3} (2 x^{3} + 1)^{\frac{3}{2}} : \frac{1}{9} (2 x^{3} + 1)^{\frac{3}{2}}

= \frac{1}{9} (2 x^{3} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (2 x^{3} + 1)^{\frac{3}{2}} + C

What is the integral of x^2*sqrt((2x^3+1)) ?
The integral of x^2*sqrt((2x^3+1)) is 1/9 (2x^3+1)^{3/2}+C