What is the integral of 8x(x^4+7) ?

The integral of 8x(x^4+7) is 8((x^6)/6+(7x^2)/2)+C

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integral of 8x(x^4+7)

\int 8 x (x^{4} + 7) d x

8 (\frac{x ^{6}}{6} + \frac{7 x ^{2}}{2}) + C

Solution steps

\int 8 x (x^{4} + 7) d x

Take the constant out:

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

= 8 \cdot \int x (x^{4} + 7) d x

Expand

x (x^{4} + 7) : x^{5} + 7 x

= 8 \cdot \int x^{5} + 7 x d x

Apply the Sum Rule:

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

= 8 (\int x^{5} d x + \int 7 x d x)

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

\int 7 x d x = \frac{7 x ^{2}}{2}

= 8 (\frac{x ^{6}}{6} + \frac{7 x ^{2}}{2})

Add a constant to the solution

= 8 (\frac{x ^{6}}{6} + \frac{7 x ^{2}}{2}) + C

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What is the integral of 8x(x^4+7) ?
The integral of 8x(x^4+7) is 8((x^6)/6+(7x^2)/2)+C