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

The integral of 2x(x^3+1) is (2x^5)/5+x^2+C

integral of 2x(x^3+1)

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

\frac{2 x ^{5}}{5} + x^{2} + C

Solution steps

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

Expand

x (x^{3} + 1) : x^{4} + x

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

\int x^{4} d x = \frac{x ^{5}}{5}

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

= 2 (\frac{x ^{5}}{5} + \frac{x ^{2}}{2})

Simplify

2 (\frac{x ^{5}}{5} + \frac{x ^{2}}{2}) : \frac{2 x ^{5}}{5} + x^{2}

= \frac{2 x ^{5}}{5} + x^{2}

Add a constant to the solution

= \frac{2 x ^{5}}{5} + x^{2} + C

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