What is the integral of 5u^5sqrt(u^6+5) ?

The integral of 5u^5sqrt(u^6+5) is 5/9 (u^6+5)^{3/2}+C

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integral of 5u^5sqrt(u^6+5)

\int 5 u^{5} u^{6} + 5 d u

\frac{5}{9} (u^{6} + 5)^{\frac{3}{2}} + C

Solution steps

\int 5 u^{5} u^{6} + 5 d u

Take the constant out:

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

= 5 \cdot \int u^{5} u^{6} + 5 d u

Apply u-substitution

= 5 \cdot \int \frac{v}{6} d v

Take the constant out:

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

= 5 \cdot \frac{1}{6} \cdot \int v d v

Apply the Power Rule

= 5 \cdot \frac{1}{6} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

v = u^{6} + 5

= 5 \cdot \frac{1}{6} \cdot \frac{2}{3} (u^{6} + 5)^{\frac{3}{2}}

Simplify

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

= \frac{5}{9} (u^{6} + 5)^{\frac{3}{2}}

Add a constant to the solution

= \frac{5}{9} (u^{6} + 5)^{\frac{3}{2}} + C

What is the integral of 5u^5sqrt(u^6+5) ?
The integral of 5u^5sqrt(u^6+5) is 5/9 (u^6+5)^{3/2}+C