What is the integral of 4x^2sqrt(3x^3+2) ?

The integral of 4x^2sqrt(3x^3+2) is 8/27 (3x^3+2)^{3/2}+C

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integral of 4x^2sqrt(3x^3+2)

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

\frac{8}{27} (3 x^{3} + 2)^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u}{9} d u

= 4 \cdot \int \frac{u}{9} d u

Take the constant out:

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

= 4 \cdot \frac{1}{9} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 4 \cdot \frac{1}{9} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 x^{3} + 2

= 4 \cdot \frac{1}{9} \cdot \frac{2}{3} (3 x^{3} + 2)^{\frac{3}{2}}

Simplify

4 \cdot \frac{1}{9} \cdot \frac{2}{3} (3 x^{3} + 2)^{\frac{3}{2}} : \frac{8}{27} (3 x^{3} + 2)^{\frac{3}{2}}

= \frac{8}{27} (3 x^{3} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{8}{27} (3 x^{3} + 2)^{\frac{3}{2}} + C

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What is the integral of 4x^2sqrt(3x^3+2) ?
The integral of 4x^2sqrt(3x^3+2) is 8/27 (3x^3+2)^{3/2}+C