What is the integral of (5x+4)(5x^2+8x)^4 ?

The integral of (5x+4)(5x^2+8x)^4 is (625x^{10})/2+2500x^9+8000x^8+12800x^7+10240x^6+(16384x^5)/5+C

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integral of (5x+4)(5x^2+8x)^4

\int (5 x + 4) (5 x^{2} + 8 x)^{4} d x

\frac{625 x ^{10}}{2} + 2500 x^{9} + 8000 x^{8} + 12800 x^{7} + 10240 x^{6} + \frac{16384 x ^{5}}{5} + C

Solution steps

\int (5 x + 4) (5 x^{2} + 8 x)^{4} d x

Expand

(5 x + 4) (5 x^{2} + 8 x)^{4} : 3125 x^{9} + 22500 x^{8} + 64000 x^{7} + 89600 x^{6} + 61440 x^{5} + 16384 x^{4}

= \int 3125 x^{9} + 22500 x^{8} + 64000 x^{7} + 89600 x^{6} + 61440 x^{5} + 16384 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 3125 x^{9} d x + \int 22500 x^{8} d x + \int 64000 x^{7} d x + \int 89600 x^{6} d x + \int 61440 x^{5} d x + \int 16384 x^{4} d x

\int 3125 x^{9} d x = \frac{625 x ^{10}}{2}

\int 22500 x^{8} d x = 2500 x^{9}

\int 64000 x^{7} d x = 8000 x^{8}

\int 89600 x^{6} d x = 12800 x^{7}

\int 61440 x^{5} d x = 10240 x^{6}

\int 16384 x^{4} d x = \frac{16384 x ^{5}}{5}

= \frac{625 x ^{10}}{2} + 2500 x^{9} + 8000 x^{8} + 12800 x^{7} + 10240 x^{6} + \frac{16384 x ^{5}}{5}

Add a constant to the solution

= \frac{625 x ^{10}}{2} + 2500 x^{9} + 8000 x^{8} + 12800 x^{7} + 10240 x^{6} + \frac{16384 x ^{5}}{5} + C

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\int (4 x^{3} + 5) x^{4} + 5 x d x

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\int \frac{3 x}{x - 2} d x

What is the integral of (5x+4)(5x^2+8x)^4 ?
The integral of (5x+4)(5x^2+8x)^4 is (625x^{10})/2+2500x^9+8000x^8+12800x^7+10240x^6+(16384x^5)/5+C