What is the integral of x(x+5)^6 ?

The integral of x(x+5)^6 is (x^8)/8+(30x^7)/7+(125x^6)/2+500x^5+(9375x^4)/4+6250x^3+(15625x^2)/2+C

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integral of x(x+5)^6

\int x (x + 5)^{6} d x

\frac{x ^{8}}{8} + \frac{30 x ^{7}}{7} + \frac{125 x ^{6}}{2} + 500 x^{5} + \frac{9375 x ^{4}}{4} + 6250 x^{3} + \frac{15625 x ^{2}}{2} + C

Solution steps

\int x (x + 5)^{6} d x

Expand

x (x + 5)^{6} : x^{7} + 30 x^{6} + 375 x^{5} + 2500 x^{4} + 9375 x^{3} + 18750 x^{2} + 15625 x

= \int x^{7} + 30 x^{6} + 375 x^{5} + 2500 x^{4} + 9375 x^{3} + 18750 x^{2} + 15625 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^{7} d x + \int 30 x^{6} d x + \int 375 x^{5} d x + \int 2500 x^{4} d x + \int 9375 x^{3} d x + \int 18750 x^{2} d x + \int 15625 x d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 30 x^{6} d x = \frac{30 x ^{7}}{7}

\int 375 x^{5} d x = \frac{125 x ^{6}}{2}

\int 2500 x^{4} d x = 500 x^{5}

\int 9375 x^{3} d x = \frac{9375 x ^{4}}{4}

\int 18750 x^{2} d x = 6250 x^{3}

\int 15625 x d x = \frac{15625 x ^{2}}{2}

= \frac{x ^{8}}{8} + \frac{30 x ^{7}}{7} + \frac{125 x ^{6}}{2} + 500 x^{5} + \frac{9375 x ^{4}}{4} + 6250 x^{3} + \frac{15625 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{8}}{8} + \frac{30 x ^{7}}{7} + \frac{125 x ^{6}}{2} + 500 x^{5} + \frac{9375 x ^{4}}{4} + 6250 x^{3} + \frac{15625 x ^{2}}{2} + C

\frac{\partial}{\partial x} (2 x y - e^{x})

\frac{\partial}{\partial y} (\frac{t ^{2}}{4 y ^{4}})

\int \frac{3 x + 1}{( x ^{2} + 2 ) ( x + 3 )} d x

\int 2 x x - 3 d x

\int \frac{1}{( y - 3 ) ( y - 2 ) ( y - 1 )} d y

What is the integral of x(x+5)^6 ?
The integral of x(x+5)^6 is (x^8)/8+(30x^7)/7+(125x^6)/2+500x^5+(9375x^4)/4+6250x^3+(15625x^2)/2+C