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

The integral of (x+2)^2(x-3)^3 is (x^6)/6-x^5-(5x^4)/4+15x^3-108x+C

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

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

\frac{x ^{6}}{6} - x^{5} - \frac{5 x ^{4}}{4} + 15 x^{3} - 108 x + C

Solution steps

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

Expand

(x + 2)^{2} (x - 3)^{3} : x^{5} - 5 x^{4} - 5 x^{3} + 45 x^{2} - 108

= \int x^{5} - 5 x^{4} - 5 x^{3} + 45 x^{2} - 108 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 5 x^{4} d x - \int 5 x^{3} d x + \int 45 x^{2} d x - \int 108 d x

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

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

\int 5 x^{3} d x = \frac{5 x ^{4}}{4}

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

\int 108 d x = 108 x

= \frac{x ^{6}}{6} - x^{5} - \frac{5 x ^{4}}{4} + 15 x^{3} - 108 x

Add a constant to the solution

= \frac{x ^{6}}{6} - x^{5} - \frac{5 x ^{4}}{4} + 15 x^{3} - 108 x + C

What is the integral of (x+2)^2(x-3)^3 ?
The integral of (x+2)^2(x-3)^3 is (x^6)/6-x^5-(5x^4)/4+15x^3-108x+C