What is the integral from 1 to 3 of 5x(x^2+1)^3 ?

The integral from 1 to 3 of 5x(x^2+1)^3 is 6240

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integral from 1 to 3 of 5x(x^2+1)^3

\int_{1}^{3} 5 x (x^{2} + 1)^{3} d x

6240

Solution steps

\int_{1}^{3} 5 x (x^{2} + 1)^{3} d x

Take the constant out:

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

= 5 \cdot \int_{1}^{3} x (x^{2} + 1)^{3} d x

Expand

x (x^{2} + 1)^{3} : x^{7} + 3 x^{5} + 3 x^{3} + x

= 5 \cdot \int_{1}^{3} x^{7} + 3 x^{5} + 3 x^{3} + 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

= 5 (\int_{1}^{3} x^{7} d x + \int_{1}^{3} 3 x^{5} d x + \int_{1}^{3} 3 x^{3} d x + \int_{1}^{3} x d x)

\int_{1}^{3} x^{7} d x = 820

\int_{1}^{3} 3 x^{5} d x = 364

\int_{1}^{3} 3 x^{3} d x = 60

\int_{1}^{3} x d x = 4

= 5 (820 + 364 + 60 + 4)

Simplify

= 6240

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What is the integral from 1 to 3 of 5x(x^2+1)^3 ?
The integral from 1 to 3 of 5x(x^2+1)^3 is 6240