What is the integral of (x+1)^3(x-2) ?

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

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integral of (x+1)^3(x-2)

\int (x + 1)^{3} (x - 2) d x

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

Solution steps

\int (x + 1)^{3} (x - 2) d x

Expand

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

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

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

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

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

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 2 d x = 2 x

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

Add a constant to the solution

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

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