What is the integral from 0 to 2 of 4x(x^2+1)^3 ?

The integral from 0 to 2 of 4x(x^2+1)^3 is 312

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integral from 0 to 2 of 4x(x^2+1)^3

\int_{0}^{2} 4 x (x^{2} + 1)^{3} d x

312

Solution steps

\int_{0}^{2} 4 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

= 4 \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

= 4 \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

= 4 (\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

= 4 (32 + 32 + 12 + 2)

Simplify

= 312

What is the integral from 0 to 2 of 4x(x^2+1)^3 ?
The integral from 0 to 2 of 4x(x^2+1)^3 is 312