What is the integral of (x^3(x^2-2x)^3) ?

The integral of (x^3(x^2-2x)^3) is (x^{10}}{10}-\frac{2x^9)/3+(3x^8)/2-(8x^7)/7+C

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integral of (x^3(x^2-2x)^3)

\int (x^{3} (x^{2} - 2 x)^{3}) d x

\frac{x ^{10}}{10} - \frac{2 x ^{9}}{3} + \frac{3 x ^{8}}{2} - \frac{8 x ^{7}}{7} + C

Solution steps

\int x^{3} (x^{2} - 2 x)^{3} d x

Expand

x^{3} (x^{2} - 2 x)^{3} : x^{9} - 6 x^{8} + 12 x^{7} - 8 x^{6}

= \int x^{9} - 6 x^{8} + 12 x^{7} - 8 x^{6} 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^{9} d x - \int 6 x^{8} d x + \int 12 x^{7} d x - \int 8 x^{6} d x

\int x^{9} d x = \frac{x ^{10}}{10}

\int 6 x^{8} d x = \frac{2 x ^{9}}{3}

\int 12 x^{7} d x = \frac{3 x ^{8}}{2}

\int 8 x^{6} d x = \frac{8 x ^{7}}{7}

= \frac{x ^{10}}{10} - \frac{2 x ^{9}}{3} + \frac{3 x ^{8}}{2} - \frac{8 x ^{7}}{7}

Add a constant to the solution

= \frac{x ^{10}}{10} - \frac{2 x ^{9}}{3} + \frac{3 x ^{8}}{2} - \frac{8 x ^{7}}{7} + C

y = \frac{1}{2} (e^{x} - e^{- x})

925 + 20 x - 0.079 x^{2}

\int_{- 9}^{0} \frac{1}{16 - x} d x

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

\int x^{2} - x d x

What is the integral of (x^3(x^2-2x)^3) ?
The integral of (x^3(x^2-2x)^3) is (x^{10}}{10}-\frac{2x^9)/3+(3x^8)/2-(8x^7)/7+C