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

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

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integral of 2x(x^2+2)2

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

x^{4} + 4 x^{2} + C

Solution steps

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

Take the constant out:

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

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

Simplify

= 4 \cdot \int x (x^{2} + 2) d x

Expand

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

= 4 \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

= 4 (\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}

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

Simplify

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

= x^{4} + 4 x^{2}

Add a constant to the solution

= x^{4} + 4 x^{2} + C