What is the integral of 8x^2sqrt(6x^3+5) ?

The integral of 8x^2sqrt(6x^3+5) is 8/27 (6x^3+5)^{3/2}+C

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integral of 8x^2sqrt(6x^3+5)

\int 8 x^{2} 6 x^{3} + 5 d x

\frac{8}{27} (6 x^{3} + 5)^{\frac{3}{2}} + C

Solution steps

\int 8 x^{2} 6 x^{3} + 5 d x

Take the constant out:

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

= 8 \cdot \int x^{2} 6 x^{3} + 5 d x

Apply u-substitution

= 8 \cdot \int \frac{u ^{2}}{9} d u

Take the constant out:

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

= 8 \cdot \frac{1}{9} \cdot \int u^{2} d u

Apply the Power Rule

= 8 \cdot \frac{1}{9} \cdot \frac{u ^{3}}{3}

Substitute back

u = 6 x^{3} + 5

= 8 \cdot \frac{1}{9} \cdot \frac{( 6 x ^{3} + 5 ) ^{3}}{3}

Simplify

8 \cdot \frac{1}{9} \cdot \frac{( 6 x ^{3} + 5 ) ^{3}}{3} : \frac{8}{27} (6 x^{3} + 5)^{\frac{3}{2}}

= \frac{8}{27} (6 x^{3} + 5)^{\frac{3}{2}}

Add a constant to the solution

= \frac{8}{27} (6 x^{3} + 5)^{\frac{3}{2}} + C

What is the integral of 8x^2sqrt(6x^3+5) ?
The integral of 8x^2sqrt(6x^3+5) is 8/27 (6x^3+5)^{3/2}+C