What is the integral of (8x-x^4)^3(8x^3-16) ?

The integral of (8x-x^4)^3(8x^3-16) is -(x^{16})/2+16x^{13}-192x^{10}+1024x^7-2048x^4+C

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integral of (8x-x^4)^3(8x^3-16)

integral

(8 x - x^{4})^{3} (8 x^{3} - 16)

- \frac{x ^{16}}{2} + 16 x^{13} - 192 x^{10} + 1024 x^{7} - 2048 x^{4} + C

Solution steps

\int (8 x - x^{4})^{3} (8 x^{3} - 16) d x

Expand

(8 x - x^{4})^{3} (8 x^{3} - 16) : - 8 x^{15} + 208 x^{12} - 1920 x^{9} + 7168 x^{6} - 8192 x^{3}

= \int - 8 x^{15} + 208 x^{12} - 1920 x^{9} + 7168 x^{6} - 8192 x^{3} 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 8 x^{15} d x + \int 208 x^{12} d x - \int 1920 x^{9} d x + \int 7168 x^{6} d x - \int 8192 x^{3} d x

\int 8 x^{15} d x = \frac{x ^{16}}{2}

\int 208 x^{12} d x = 16 x^{13}

\int 1920 x^{9} d x = 192 x^{10}

\int 7168 x^{6} d x = 1024 x^{7}

\int 8192 x^{3} d x = 2048 x^{4}

= - \frac{x ^{16}}{2} + 16 x^{13} - 192 x^{10} + 1024 x^{7} - 2048 x^{4}

Add a constant to the solution

= - \frac{x ^{16}}{2} + 16 x^{13} - 192 x^{10} + 1024 x^{7} - 2048 x^{4} + C

cos^{2} (\frac{x}{4})

(\frac{1}{x ^{4}}) sin (x)

cos ((\frac{1}{4}) x)

(t^{2} - 6 t + 13)^{- 1}

\frac{( 1 + x ) ^{3}}{x}

What is the integral of (8x-x^4)^3(8x^3-16) ?
The integral of (8x-x^4)^3(8x^3-16) is -(x^{16})/2+16x^{13}-192x^{10}+1024x^7-2048x^4+C