What is the integral of x^4(x-2)(x+3) ?

The integral of x^4(x-2)(x+3) is (x^7)/7+(x^6)/6-(6x^5)/5+C

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integral of x^4(x-2)(x+3)

\int x^{4} (x - 2) (x + 3) d x

\frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{6 x ^{5}}{5} + C

Solution steps

\int x^{4} (x - 2) (x + 3) d x

Expand

x^{4} (x - 2) (x + 3) : x^{6} + x^{5} - 6 x^{4}

= \int x^{6} + x^{5} - 6 x^{4} 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^{6} d x + \int x^{5} d x - \int 6 x^{4} d x

\int x^{6} d x = \frac{x ^{7}}{7}

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

\int 6 x^{4} d x = \frac{6 x ^{5}}{5}

= \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{6 x ^{5}}{5}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{x ^{6}}{6} - \frac{6 x ^{5}}{5} + C

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What is the integral of x^4(x-2)(x+3) ?
The integral of x^4(x-2)(x+3) is (x^7)/7+(x^6)/6-(6x^5)/5+C