What is the integral of 2x(x^2+2) ?

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

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integral of 2x(x^2+2)

\int 2 x (x^{2} + 2) d x

2 (\frac{x ^{4}}{4} + x^{2}) + C

Solution steps

\int 2 x (x^{2} + 2) d x

Take the constant out:

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

= 2 \cdot \int x (x^{2} + 2) d x

Expand

x (x^{2} + 2) : x^{3} + 2 x

= 2 \cdot \int x^{3} + 2 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 x^{3} d x + \int 2 x d x)

\int x^{3} d x = \frac{x ^{4}}{4}

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

= 2 (\frac{x ^{4}}{4} + x^{2})

Add a constant to the solution

= 2 (\frac{x ^{4}}{4} + x^{2}) + C

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