What is the integral of (x+2)(x-5)^4 ?

The integral of (x+2)(x-5)^4 is (x^6)/6-(18x^5)/5+(55x^4)/2-(200x^3)/3-(375x^2)/2+1250x+C

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integral of (x+2)(x-5)^4

\int (x + 2) (x - 5)^{4} d x

\frac{x ^{6}}{6} - \frac{18 x ^{5}}{5} + \frac{55 x ^{4}}{2} - \frac{200 x ^{3}}{3} - \frac{375 x ^{2}}{2} + 1250 x + C

Solution steps

\int (x + 2) (x - 5)^{4} d x

Expand

(x + 2) (x - 5)^{4} : x^{5} - 18 x^{4} + 110 x^{3} - 200 x^{2} - 375 x + 1250

= \int x^{5} - 18 x^{4} + 110 x^{3} - 200 x^{2} - 375 x + 1250 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^{5} d x - \int 18 x^{4} d x + \int 110 x^{3} d x - \int 200 x^{2} d x - \int 375 x d x + \int 1250 d x

\int x^{5} d x = \frac{x ^{6}}{6}

\int 18 x^{4} d x = \frac{18 x ^{5}}{5}

\int 110 x^{3} d x = \frac{55 x ^{4}}{2}

\int 200 x^{2} d x = \frac{200 x ^{3}}{3}

\int 375 x d x = \frac{375 x ^{2}}{2}

\int 1250 d x = 1250 x

= \frac{x ^{6}}{6} - \frac{18 x ^{5}}{5} + \frac{55 x ^{4}}{2} - \frac{200 x ^{3}}{3} - \frac{375 x ^{2}}{2} + 1250 x

Add a constant to the solution

= \frac{x ^{6}}{6} - \frac{18 x ^{5}}{5} + \frac{55 x ^{4}}{2} - \frac{200 x ^{3}}{3} - \frac{375 x ^{2}}{2} + 1250 x + C

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What is the integral of (x+2)(x-5)^4 ?
The integral of (x+2)(x-5)^4 is (x^6)/6-(18x^5)/5+(55x^4)/2-(200x^3)/3-(375x^2)/2+1250x+C