What is the integral of x^2(4x^3-9)^5 ?

The integral of x^2(4x^3-9)^5 is (512x^{18})/9-768x^{15}+4320x^{12}-12960x^9+21870x^6-19683x^3+C

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integral of x^2(4x^3-9)^5

\int x^{2} (4 x^{3} - 9)^{5} d x

\frac{512 x ^{18}}{9} - 768 x^{15} + 4320 x^{12} - 12960 x^{9} + 21870 x^{6} - 19683 x^{3} + C

Solution steps

\int x^{2} (4 x^{3} - 9)^{5} d x

Expand

x^{2} (4 x^{3} - 9)^{5} : 1024 x^{17} - 11520 x^{14} + 51840 x^{11} - 116640 x^{8} + 131220 x^{5} - 59049 x^{2}

= \int 1024 x^{17} - 11520 x^{14} + 51840 x^{11} - 116640 x^{8} + 131220 x^{5} - 59049 x^{2} 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 1024 x^{17} d x - \int 11520 x^{14} d x + \int 51840 x^{11} d x - \int 116640 x^{8} d x + \int 131220 x^{5} d x - \int 59049 x^{2} d x

\int 1024 x^{17} d x = \frac{512 x ^{18}}{9}

\int 11520 x^{14} d x = 768 x^{15}

\int 51840 x^{11} d x = 4320 x^{12}

\int 116640 x^{8} d x = 12960 x^{9}

\int 131220 x^{5} d x = 21870 x^{6}

\int 59049 x^{2} d x = 19683 x^{3}

= \frac{512 x ^{18}}{9} - 768 x^{15} + 4320 x^{12} - 12960 x^{9} + 21870 x^{6} - 19683 x^{3}

Add a constant to the solution

= \frac{512 x ^{18}}{9} - 768 x^{15} + 4320 x^{12} - 12960 x^{9} + 21870 x^{6} - 19683 x^{3} + C

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What is the integral of x^2(4x^3-9)^5 ?
The integral of x^2(4x^3-9)^5 is (512x^{18})/9-768x^{15}+4320x^{12}-12960x^9+21870x^6-19683x^3+C