What is the integral from 0 to 2 of 2x(x^2+1)^3 ?

The integral from 0 to 2 of 2x(x^2+1)^3 is 156

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integral from 0 to 2 of 2x(x^2+1)^3

\int_{0}^{2} 2 x (x^{2} + 1)^{3} d x

156

Solution steps

\int_{0}^{2} 2 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

= 2 \cdot \int_{0}^{2} x (x^{2} + 1)^{3} d x

Expand

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

= 2 \cdot \int_{0}^{2} 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

= 2 (\int_{0}^{2} x^{7} d x + \int_{0}^{2} 3 x^{5} d x + \int_{0}^{2} 3 x^{3} d x + \int_{0}^{2} x d x)

\int_{0}^{2} x^{7} d x = 32

\int_{0}^{2} 3 x^{5} d x = 32

\int_{0}^{2} 3 x^{3} d x = 12

\int_{0}^{2} x d x = 2

= 2 (32 + 32 + 12 + 2)

Simplify

= 156

\int_{- 2}^{- 1} x^{2} d x

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\int_{1}^{7} x^{2} - 2 x - 24 d x

What is the integral from 0 to 2 of 2x(x^2+1)^3 ?
The integral from 0 to 2 of 2x(x^2+1)^3 is 156