What is the integral of x(5x^2+2/3)^4 ?

The integral of x(5x^2+2/3)^4 is (125x^{10})/2+(125x^8)/3+(100x^6)/9+(40x^4)/(27)+(8x^2)/(81)+C

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integral of x(5x^2+2/3)^4

\int x (5 x^{2} + \frac{2}{3})^{4} d x

\frac{125 x ^{10}}{2} + \frac{125 x ^{8}}{3} + \frac{100 x ^{6}}{9} + \frac{40 x ^{4}}{27} + \frac{8 x ^{2}}{81} + C

Solution steps

\int x (5 x^{2} + \frac{2}{3})^{4} d x

Expand

x (5 x^{2} + \frac{2}{3})^{4} : 625 x^{9} + \frac{1000 x ^{7}}{3} + \frac{200 x ^{5}}{3} + \frac{160 x ^{3}}{27} + \frac{16 x}{81}

= \int 625 x^{9} + \frac{1000 x ^{7}}{3} + \frac{200 x ^{5}}{3} + \frac{160 x ^{3}}{27} + \frac{16 x}{81} 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 625 x^{9} d x + \int \frac{1000 x ^{7}}{3} d x + \int \frac{200 x ^{5}}{3} d x + \int \frac{160 x ^{3}}{27} d x + \int \frac{16 x}{81} d x

\int 625 x^{9} d x = \frac{125 x ^{10}}{2}

\int \frac{1000 x ^{7}}{3} d x = \frac{125 x ^{8}}{3}

\int \frac{200 x ^{5}}{3} d x = \frac{100 x ^{6}}{9}

\int \frac{160 x ^{3}}{27} d x = \frac{40 x ^{4}}{27}

\int \frac{16 x}{81} d x = \frac{8 x ^{2}}{81}

= \frac{125 x ^{10}}{2} + \frac{125 x ^{8}}{3} + \frac{100 x ^{6}}{9} + \frac{40 x ^{4}}{27} + \frac{8 x ^{2}}{81}

Add a constant to the solution

= \frac{125 x ^{10}}{2} + \frac{125 x ^{8}}{3} + \frac{100 x ^{6}}{9} + \frac{40 x ^{4}}{27} + \frac{8 x ^{2}}{81} + C

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What is the integral of x(5x^2+2/3)^4 ?
The integral of x(5x^2+2/3)^4 is (125x^{10})/2+(125x^8)/3+(100x^6)/9+(40x^4)/(27)+(8x^2)/(81)+C