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

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

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integral of 2x(x+1)

\int 2 x (x + 1) d x

\frac{2 x ^{3}}{3} + x^{2} + C

Solution steps

\int 2 x (x + 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 + 1) d x

Expand

x (x + 1) : x^{2} + x

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

\int x^{2} d x = \frac{x ^{3}}{3}

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

= 2 (\frac{x ^{3}}{3} + \frac{x ^{2}}{2})

Simplify

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

= \frac{2 x ^{3}}{3} + x^{2}

Add a constant to the solution

= \frac{2 x ^{3}}{3} + x^{2} + C

x \to 2 lim (x^{3} + 8)

\frac{d}{d x} (7 x arcsin (x))

\frac{200 t}{t + 2}

\int \frac{1}{( ln ( x ) ) ^{3} x} d x

\frac{d}{d x} (x^{5} - 2 x^{3} + x - 1)