What is the integral of (5x^3+10x)^3(3x^2+2) ?

The integral of (5x^3+10x)^3(3x^2+2) is (125x^{12})/4+250x^{10}+750x^8+1000x^6+500x^4+C

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integral of (5x^3+10x)^3(3x^2+2)

\int (5 x^{3} + 10 x)^{3} (3 x^{2} + 2) d x

\frac{125 x ^{12}}{4} + 250 x^{10} + 750 x^{8} + 1000 x^{6} + 500 x^{4} + C

Solution steps

\int (5 x^{3} + 10 x)^{3} (3 x^{2} + 2) d x

Expand

(5 x^{3} + 10 x)^{3} (3 x^{2} + 2) : 375 x^{11} + 2500 x^{9} + 6000 x^{7} + 6000 x^{5} + 2000 x^{3}

= \int 375 x^{11} + 2500 x^{9} + 6000 x^{7} + 6000 x^{5} + 2000 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 375 x^{11} d x + \int 2500 x^{9} d x + \int 6000 x^{7} d x + \int 6000 x^{5} d x + \int 2000 x^{3} d x

\int 375 x^{11} d x = \frac{125 x ^{12}}{4}

\int 2500 x^{9} d x = 250 x^{10}

\int 6000 x^{7} d x = 750 x^{8}

\int 6000 x^{5} d x = 1000 x^{6}

\int 2000 x^{3} d x = 500 x^{4}

= \frac{125 x ^{12}}{4} + 250 x^{10} + 750 x^{8} + 1000 x^{6} + 500 x^{4}

Add a constant to the solution

= \frac{125 x ^{12}}{4} + 250 x^{10} + 750 x^{8} + 1000 x^{6} + 500 x^{4} + C

x^{5} + 5^{x}

\int_{0}^{1} sin (x) d x

x \to 0 lim (\frac{2 x ^{2}}{sin ^{2} ( 8 x )})

f (x) = x^{4} - 2 x^{3} - 5 x - 3, at x = 1

\frac{1}{16 x ^{2}}, x, \frac{x}{10}

What is the integral of (5x^3+10x)^3(3x^2+2) ?
The integral of (5x^3+10x)^3(3x^2+2) is (125x^{12})/4+250x^{10}+750x^8+1000x^6+500x^4+C