What is the integral of 5v(v^2+8)^2 ?

The integral of 5v(v^2+8)^2 is 5((v^6)/6+4v^4+32v^2)+C

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integral of 5v(v^2+8)^2

\int 5 v (v^{2} + 8)^{2} d v

5 (\frac{v ^{6}}{6} + 4 v^{4} + 32 v^{2}) + C

Solution steps

\int 5 v (v^{2} + 8)^{2} d v

Take the constant out:

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

= 5 \cdot \int v (v^{2} + 8)^{2} d v

Expand

v (v^{2} + 8)^{2} : v^{5} + 16 v^{3} + 64 v

= 5 \cdot \int v^{5} + 16 v^{3} + 64 v d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (\int v^{5} d v + \int 16 v^{3} d v + \int 64 v d v)

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

\int 16 v^{3} d v = 4 v^{4}

\int 64 v d v = 32 v^{2}

= 5 (\frac{v ^{6}}{6} + 4 v^{4} + 32 v^{2})

Add a constant to the solution

= 5 (\frac{v ^{6}}{6} + 4 v^{4} + 32 v^{2}) + C

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What is the integral of 5v(v^2+8)^2 ?
The integral of 5v(v^2+8)^2 is 5((v^6)/6+4v^4+32v^2)+C