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

The integral of (x+2)(x-3)^3 is (x^5)/5-(7x^4)/4+3x^3+(27x^2)/2-54x+C

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

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

\frac{x ^{5}}{5} - \frac{7 x ^{4}}{4} + 3 x^{3} + \frac{27 x ^{2}}{2} - 54 x + C

Solution steps

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

Expand

(x + 2) (x - 3)^{3} : x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54

= \int x^{4} - 7 x^{3} + 9 x^{2} + 27 x - 54 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 7 x^{3} d x + \int 9 x^{2} d x + \int 27 x d x - \int 54 d x

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

\int 7 x^{3} d x = \frac{7 x ^{4}}{4}

\int 9 x^{2} d x = 3 x^{3}

\int 27 x d x = \frac{27 x ^{2}}{2}

\int 54 d x = 54 x

= \frac{x ^{5}}{5} - \frac{7 x ^{4}}{4} + 3 x^{3} + \frac{27 x ^{2}}{2} - 54 x

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{7 x ^{4}}{4} + 3 x^{3} + \frac{27 x ^{2}}{2} - 54 x + C

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What is the integral of (x+2)(x-3)^3 ?
The integral of (x+2)(x-3)^3 is (x^5)/5-(7x^4)/4+3x^3+(27x^2)/2-54x+C