What is the integral of x(x+6)^4 ?

The integral of x(x+6)^4 is (x^6)/6+(24x^5)/5+54x^4+288x^3+648x^2+C

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integral of x(x+6)^4

\int x (x + 6)^{4} d x

\frac{x ^{6}}{6} + \frac{24 x ^{5}}{5} + 54 x^{4} + 288 x^{3} + 648 x^{2} + C

Solution steps

\int x (x + 6)^{4} d x

Expand

x (x + 6)^{4} : x^{5} + 24 x^{4} + 216 x^{3} + 864 x^{2} + 1296 x

= \int x^{5} + 24 x^{4} + 216 x^{3} + 864 x^{2} + 1296 x 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 24 x^{4} d x + \int 216 x^{3} d x + \int 864 x^{2} d x + \int 1296 x d x

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

\int 24 x^{4} d x = \frac{24 x ^{5}}{5}

\int 216 x^{3} d x = 54 x^{4}

\int 864 x^{2} d x = 288 x^{3}

\int 1296 x d x = 648 x^{2}

= \frac{x ^{6}}{6} + \frac{24 x ^{5}}{5} + 54 x^{4} + 288 x^{3} + 648 x^{2}

Add a constant to the solution

= \frac{x ^{6}}{6} + \frac{24 x ^{5}}{5} + 54 x^{4} + 288 x^{3} + 648 x^{2} + C

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What is the integral of x(x+6)^4 ?
The integral of x(x+6)^4 is (x^6)/6+(24x^5)/5+54x^4+288x^3+648x^2+C