What is the integral of (4x^3+12x)^3(3x^2+3) ?

The integral of (4x^3+12x)^3(3x^2+3) is 16x^{12}+192x^{10}+864x^8+1728x^6+1296x^4+C

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integral of (4x^3+12x)^3(3x^2+3)

\int (4 x^{3} + 12 x)^{3} (3 x^{2} + 3) d x

16 x^{12} + 192 x^{10} + 864 x^{8} + 1728 x^{6} + 1296 x^{4} + C

Solution steps

\int (4 x^{3} + 12 x)^{3} (3 x^{2} + 3) d x

Expand

(4 x^{3} + 12 x)^{3} (3 x^{2} + 3) : 192 x^{11} + 1920 x^{9} + 6912 x^{7} + 10368 x^{5} + 5184 x^{3}

= \int 192 x^{11} + 1920 x^{9} + 6912 x^{7} + 10368 x^{5} + 5184 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 192 x^{11} d x + \int 1920 x^{9} d x + \int 6912 x^{7} d x + \int 10368 x^{5} d x + \int 5184 x^{3} d x

\int 192 x^{11} d x = 16 x^{12}

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

\int 6912 x^{7} d x = 864 x^{8}

\int 10368 x^{5} d x = 1728 x^{6}

\int 5184 x^{3} d x = 1296 x^{4}

= 16 x^{12} + 192 x^{10} + 864 x^{8} + 1728 x^{6} + 1296 x^{4}

Add a constant to the solution

= 16 x^{12} + 192 x^{10} + 864 x^{8} + 1728 x^{6} + 1296 x^{4} + C

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What is the integral of (4x^3+12x)^3(3x^2+3) ?
The integral of (4x^3+12x)^3(3x^2+3) is 16x^{12}+192x^{10}+864x^8+1728x^6+1296x^4+C