What is the integral of x(x+2)^5 ?

The integral of x(x+2)^5 is (x^7)/7+(5x^6)/3+8x^5+20x^4+(80x^3)/3+16x^2+C

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integral of x(x+2)^5

\int x (x + 2)^{5} d x

\frac{x ^{7}}{7} + \frac{5 x ^{6}}{3} + 8 x^{5} + 20 x^{4} + \frac{80 x ^{3}}{3} + 16 x^{2} + C

Solution steps

\int x (x + 2)^{5} d x

Expand

x (x + 2)^{5} : x^{6} + 10 x^{5} + 40 x^{4} + 80 x^{3} + 80 x^{2} + 32 x

= \int x^{6} + 10 x^{5} + 40 x^{4} + 80 x^{3} + 80 x^{2} + 32 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^{6} d x + \int 10 x^{5} d x + \int 40 x^{4} d x + \int 80 x^{3} d x + \int 80 x^{2} d x + \int 32 x d x

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

\int 10 x^{5} d x = \frac{5 x ^{6}}{3}

\int 40 x^{4} d x = 8 x^{5}

\int 80 x^{3} d x = 20 x^{4}

\int 80 x^{2} d x = \frac{80 x ^{3}}{3}

\int 32 x d x = 16 x^{2}

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{3} + 8 x^{5} + 20 x^{4} + \frac{80 x ^{3}}{3} + 16 x^{2}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{3} + 8 x^{5} + 20 x^{4} + \frac{80 x ^{3}}{3} + 16 x^{2} + C

What is the integral of x(x+2)^5 ?
The integral of x(x+2)^5 is (x^7)/7+(5x^6)/3+8x^5+20x^4+(80x^3)/3+16x^2+C