What is the integral of (x+2)(x-5)^3 ?

The integral of (x+2)(x-5)^3 is (x^5)/5-(13x^4)/4+15x^3+(25x^2)/2-250x+C

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integral of (x+2)(x-5)^3

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

\frac{x ^{5}}{5} - \frac{13 x ^{4}}{4} + 15 x^{3} + \frac{25 x ^{2}}{2} - 250 x + C

Solution steps

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

Expand

(x + 2) (x - 5)^{3} : x^{4} - 13 x^{3} + 45 x^{2} + 25 x - 250

= \int x^{4} - 13 x^{3} + 45 x^{2} + 25 x - 250 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 13 x^{3} d x + \int 45 x^{2} d x + \int 25 x d x - \int 250 d x

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

\int 13 x^{3} d x = \frac{13 x ^{4}}{4}

\int 45 x^{2} d x = 15 x^{3}

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

\int 250 d x = 250 x

= \frac{x ^{5}}{5} - \frac{13 x ^{4}}{4} + 15 x^{3} + \frac{25 x ^{2}}{2} - 250 x

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{13 x ^{4}}{4} + 15 x^{3} + \frac{25 x ^{2}}{2} - 250 x + C

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What is the integral of (x+2)(x-5)^3 ?
The integral of (x+2)(x-5)^3 is (x^5)/5-(13x^4)/4+15x^3+(25x^2)/2-250x+C