What is the integral of x^2(1-2x^3)^5 ?

The integral of x^2(1-2x^3)^5 is (x^3)/3-(5x^6)/3+(40x^9)/9-(20x^{12})/3+(16x^{15})/3-(16x^{18})/9+C

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integral of x^2(1-2x^3)^5

\int x^{2} (1 - 2 x^{3})^{5} d x

\frac{x ^{3}}{3} - \frac{5 x ^{6}}{3} + \frac{40 x ^{9}}{9} - \frac{20 x ^{12}}{3} + \frac{16 x ^{15}}{3} - \frac{16 x ^{18}}{9} + C

Solution steps

\int x^{2} (1 - 2 x^{3})^{5} d x

Expand

x^{2} (1 - 2 x^{3})^{5} : x^{2} - 10 x^{5} + 40 x^{8} - 80 x^{11} + 80 x^{14} - 32 x^{17}

= \int x^{2} - 10 x^{5} + 40 x^{8} - 80 x^{11} + 80 x^{14} - 32 x^{17} 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^{2} d x - \int 10 x^{5} d x + \int 40 x^{8} d x - \int 80 x^{11} d x + \int 80 x^{14} d x - \int 32 x^{17} d x

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

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

\int 40 x^{8} d x = \frac{40 x ^{9}}{9}

\int 80 x^{11} d x = \frac{20 x ^{12}}{3}

\int 80 x^{14} d x = \frac{16 x ^{15}}{3}

\int 32 x^{17} d x = \frac{16 x ^{18}}{9}

= \frac{x ^{3}}{3} - \frac{5 x ^{6}}{3} + \frac{40 x ^{9}}{9} - \frac{20 x ^{12}}{3} + \frac{16 x ^{15}}{3} - \frac{16 x ^{18}}{9}

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{5 x ^{6}}{3} + \frac{40 x ^{9}}{9} - \frac{20 x ^{12}}{3} + \frac{16 x ^{15}}{3} - \frac{16 x ^{18}}{9} + C

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\int tan (z) d z

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What is the integral of x^2(1-2x^3)^5 ?
The integral of x^2(1-2x^3)^5 is (x^3)/3-(5x^6)/3+(40x^9)/9-(20x^{12})/3+(16x^{15})/3-(16x^{18})/9+C