What is the integral of (8x^3sqrt(2x^4+6)) ?

The integral of (8x^3sqrt(2x^4+6)) is 2/3 (2x^4+6)^{3/2}+C

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integral of (8x^3sqrt(2x^4+6))

\int (8 x^{3} 2 x^{4} + 6) d x

\frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}} + C

Solution steps

\int 8 x^{3} 2 x^{4} + 6 d x

Take the constant out:

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

= 8 \cdot \int x^{3} 2 x^{4} + 6 d x

Apply u-substitution

: \int \frac{u}{8} d u

= 8 \cdot \int \frac{u}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int u d u

Apply the Power Rule

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

= 8 \cdot \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 2 x^{4} + 6

= 8 \cdot \frac{1}{8} \cdot \frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}}

Simplify

8 \cdot \frac{1}{8} \cdot \frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}} : \frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}}

= \frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (2 x^{4} + 6)^{\frac{3}{2}} + C

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What is the integral of (8x^3sqrt(2x^4+6)) ?
The integral of (8x^3sqrt(2x^4+6)) is 2/3 (2x^4+6)^{3/2}+C