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

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

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integral of (x(x+1))/2

integral

\frac{x ( x + 1 )}{2}

\frac{1}{2} (\frac{x ^{3}}{3} + \frac{x ^{2}}{2}) + C

Solution steps

\int \frac{x ( x + 1 )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x (x + 1) d x

Expand

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

= \frac{1}{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

= \frac{1}{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}

= \frac{1}{2} (\frac{x ^{3}}{3} + \frac{x ^{2}}{2})

Add a constant to the solution

= \frac{1}{2} (\frac{x ^{3}}{3} + \frac{x ^{2}}{2}) + C

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