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

The integral of x(x+4)^4 is (x^6)/6+(16x^5)/5+24x^4+(256x^3)/3+128x^2+C

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integral of x(x+4)^4

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

\frac{x ^{6}}{6} + \frac{16 x ^{5}}{5} + 24 x^{4} + \frac{256 x ^{3}}{3} + 128 x^{2} + C

Solution steps

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

Expand

x (x + 4)^{4} : x^{5} + 16 x^{4} + 96 x^{3} + 256 x^{2} + 256 x

= \int x^{5} + 16 x^{4} + 96 x^{3} + 256 x^{2} + 256 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 16 x^{4} d x + \int 96 x^{3} d x + \int 256 x^{2} d x + \int 256 x d x

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

\int 16 x^{4} d x = \frac{16 x ^{5}}{5}

\int 96 x^{3} d x = 24 x^{4}

\int 256 x^{2} d x = \frac{256 x ^{3}}{3}

\int 256 x d x = 128 x^{2}

= \frac{x ^{6}}{6} + \frac{16 x ^{5}}{5} + 24 x^{4} + \frac{256 x ^{3}}{3} + 128 x^{2}

Add a constant to the solution

= \frac{x ^{6}}{6} + \frac{16 x ^{5}}{5} + 24 x^{4} + \frac{256 x ^{3}}{3} + 128 x^{2} + C

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What is the integral of x(x+4)^4 ?
The integral of x(x+4)^4 is (x^6)/6+(16x^5)/5+24x^4+(256x^3)/3+128x^2+C