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

The integral of x(x+3)(x+1) is (x^4)/4+(4x^3)/3+(3x^2)/2+C

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integral of x(x+3)(x+1)

\int x (x + 3) (x + 1) d x

\frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + \frac{3 x ^{2}}{2} + C

Solution steps

\int x (x + 3) (x + 1) d x

Expand

x (x + 3) (x + 1) : x^{3} + 4 x^{2} + 3 x

= \int x^{3} + 4 x^{2} + 3 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

= \int x^{3} d x + \int 4 x^{2} d x + \int 3 x d x

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

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

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

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + \frac{3 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + \frac{3 x ^{2}}{2} + C

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What is the integral of x(x+3)(x+1) ?
The integral of x(x+3)(x+1) is (x^4)/4+(4x^3)/3+(3x^2)/2+C