What is the integral of x*(x+5)^4 ?

The integral of x*(x+5)^4 is (x^6)/6+4x^5+(75x^4)/2+(500x^3)/3+(625x^2)/2+C

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integral of x*(x+5)^4

\int x \cdot (x + 5)^{4} d x

\frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2} + C

Solution steps

\int x (x + 5)^{4} d x

Expand

x (x + 5)^{4} : x^{5} + 20 x^{4} + 150 x^{3} + 500 x^{2} + 625 x

= \int x^{5} + 20 x^{4} + 150 x^{3} + 500 x^{2} + 625 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 20 x^{4} d x + \int 150 x^{3} d x + \int 500 x^{2} d x + \int 625 x d x

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

\int 20 x^{4} d x = 4 x^{5}

\int 150 x^{3} d x = \frac{75 x ^{4}}{2}

\int 500 x^{2} d x = \frac{500 x ^{3}}{3}

\int 625 x d x = \frac{625 x ^{2}}{2}

= \frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{6}}{6} + 4 x^{5} + \frac{75 x ^{4}}{2} + \frac{500 x ^{3}}{3} + \frac{625 x ^{2}}{2} + C

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What is the integral of x*(x+5)^4 ?
The integral of x*(x+5)^4 is (x^6)/6+4x^5+(75x^4)/2+(500x^3)/3+(625x^2)/2+C