What is the integral of x(x-5)^2(x+1) ?

The integral of x(x-5)^2(x+1) is (x^5)/5-(9x^4)/4+5x^3+(25x^2)/2+C

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integral of x(x-5)^2(x+1)

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

\frac{x ^{5}}{5} - \frac{9 x ^{4}}{4} + 5 x^{3} + \frac{25 x ^{2}}{2} + C

Solution steps

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

Expand

x (x - 5)^{2} (x + 1) : x^{4} - 9 x^{3} + 15 x^{2} + 25 x

= \int x^{4} - 9 x^{3} + 15 x^{2} + 25 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^{4} d x - \int 9 x^{3} d x + \int 15 x^{2} d x + \int 25 x d x

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

\int 9 x^{3} d x = \frac{9 x ^{4}}{4}

\int 15 x^{2} d x = 5 x^{3}

\int 25 x d x = \frac{25 x ^{2}}{2}

= \frac{x ^{5}}{5} - \frac{9 x ^{4}}{4} + 5 x^{3} + \frac{25 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{9 x ^{4}}{4} + 5 x^{3} + \frac{25 x ^{2}}{2} + C

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What is the integral of x(x-5)^2(x+1) ?
The integral of x(x-5)^2(x+1) is (x^5)/5-(9x^4)/4+5x^3+(25x^2)/2+C