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

The integral of (x^5-9x)^2(5x^4-9) is (x^{15})/3-9x^{11}+81x^7-243x^3+C

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

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

\frac{x ^{15}}{3} - 9 x^{11} + 81 x^{7} - 243 x^{3} + C

Solution steps

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

Expand

(x^{5} - 9 x)^{2} (5 x^{4} - 9) : 5 x^{14} - 99 x^{10} + 567 x^{6} - 729 x^{2}

= \int 5 x^{14} - 99 x^{10} + 567 x^{6} - 729 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 5 x^{14} d x - \int 99 x^{10} d x + \int 567 x^{6} d x - \int 729 x^{2} d x

\int 5 x^{14} d x = \frac{x ^{15}}{3}

\int 99 x^{10} d x = 9 x^{11}

\int 567 x^{6} d x = 81 x^{7}

\int 729 x^{2} d x = 243 x^{3}

= \frac{x ^{15}}{3} - 9 x^{11} + 81 x^{7} - 243 x^{3}

Add a constant to the solution

= \frac{x ^{15}}{3} - 9 x^{11} + 81 x^{7} - 243 x^{3} + C

What is the integral of (x^5-9x)^2(5x^4-9) ?
The integral of (x^5-9x)^2(5x^4-9) is (x^{15})/3-9x^{11}+81x^7-243x^3+C