What is the integral of 6u^6sqrt(u^7+8) ?

The integral of 6u^6sqrt(u^7+8) is 4/7 (u^7+8)^{3/2}+C

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integral of 6u^6sqrt(u^7+8)

\int 6 u^{6} u^{7} + 8 d u

\frac{4}{7} (u^{7} + 8)^{\frac{3}{2}} + C

Solution steps

\int 6 u^{6} u^{7} + 8 d u

Take the constant out:

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

= 6 \cdot \int u^{6} u^{7} + 8 d u

Apply u-substitution

: \int \frac{v}{7} d v

= 6 \cdot \int \frac{v}{7} d v

Take the constant out:

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

= 6 \cdot \frac{1}{7} \cdot \int v d v

Apply the Power Rule

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

= 6 \cdot \frac{1}{7} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

v = u^{7} + 8

= 6 \cdot \frac{1}{7} \cdot \frac{2}{3} (u^{7} + 8)^{\frac{3}{2}}

Simplify

6 \cdot \frac{1}{7} \cdot \frac{2}{3} (u^{7} + 8)^{\frac{3}{2}} : \frac{4}{7} (u^{7} + 8)^{\frac{3}{2}}

= \frac{4}{7} (u^{7} + 8)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{7} (u^{7} + 8)^{\frac{3}{2}} + C

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What is the integral of 6u^6sqrt(u^7+8) ?
The integral of 6u^6sqrt(u^7+8) is 4/7 (u^7+8)^{3/2}+C