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

The integral of x^4(2x^5-5)^4 is (16x^{25})/(25)-8x^{20}+40x^{15}-100x^{10}+125x^5+C

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

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

\frac{16 x ^{25}}{25} - 8 x^{20} + 40 x^{15} - 100 x^{10} + 125 x^{5} + C

Solution steps

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

Expand

x^{4} (2 x^{5} - 5)^{4} : 16 x^{24} - 160 x^{19} + 600 x^{14} - 1000 x^{9} + 625 x^{4}

= \int 16 x^{24} - 160 x^{19} + 600 x^{14} - 1000 x^{9} + 625 x^{4} 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 16 x^{24} d x - \int 160 x^{19} d x + \int 600 x^{14} d x - \int 1000 x^{9} d x + \int 625 x^{4} d x

\int 16 x^{24} d x = \frac{16 x ^{25}}{25}

\int 160 x^{19} d x = 8 x^{20}

\int 600 x^{14} d x = 40 x^{15}

\int 1000 x^{9} d x = 100 x^{10}

\int 625 x^{4} d x = 125 x^{5}

= \frac{16 x ^{25}}{25} - 8 x^{20} + 40 x^{15} - 100 x^{10} + 125 x^{5}

Add a constant to the solution

= \frac{16 x ^{25}}{25} - 8 x^{20} + 40 x^{15} - 100 x^{10} + 125 x^{5} + C

What is the integral of x^4(2x^5-5)^4 ?
The integral of x^4(2x^5-5)^4 is (16x^{25})/(25)-8x^{20}+40x^{15}-100x^{10}+125x^5+C