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

The integral from 1 to 3 of (x^2+1)^3x is 1248

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

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

1248

Solution steps

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

Expand

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

= \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

= \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

= 820 + 364 + 60 + 4

Simplify

= 1248

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What is the integral from 1 to 3 of (x^2+1)^3x ?
The integral from 1 to 3 of (x^2+1)^3x is 1248