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Popular >

integral of x(3x+2)^{12}

Solution

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

Solution

\frac{531441 x ^{14}}{14} + \frac{4251528 x ^{13}}{13} + 1299078 x^{12} + 3149280 x^{11} + 5196312 x^{10} + 6158592 x^{9} + 5388768 x^{8} + \frac{24634368 x ^{7}}{7} + 1710720 x^{6} + 608256 x^{5} + 152064 x^{4} + 24576 x^{3} + 2048 x^{2} + C

Solution steps

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

Expand

x (3 x + 2)^{12} : 531441 x^{13} + 4251528 x^{12} + 15588936 x^{11} + 34642080 x^{10} + 51963120 x^{9} + 55427328 x^{8} + 43110144 x^{7} + 24634368 x^{6} + 10264320 x^{5} + 3041280 x^{4} + 608256 x^{3} + 73728 x^{2} + 4096 x

= \int 531441 x^{13} + 4251528 x^{12} + 15588936 x^{11} + 34642080 x^{10} + 51963120 x^{9} + 55427328 x^{8} + 43110144 x^{7} + 24634368 x^{6} + 10264320 x^{5} + 3041280 x^{4} + 608256 x^{3} + 73728 x^{2} + 4096 x 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 531441 x^{13} d x + \int 4251528 x^{12} d x + \int 15588936 x^{11} d x + \int 34642080 x^{10} d x + \int 51963120 x^{9} d x + \int 55427328 x^{8} d x + \int 43110144 x^{7} d x + \int 24634368 x^{6} d x + \int 10264320 x^{5} d x + \int 3041280 x^{4} d x + \int 608256 x^{3} d x + \int 73728 x^{2} d x + \int 4096 x d x

\int 531441 x^{13} d x = \frac{531441 x ^{14}}{14}

\int 4251528 x^{12} d x = \frac{4251528 x ^{13}}{13}

\int 15588936 x^{11} d x = 1299078 x^{12}

\int 34642080 x^{10} d x = 3149280 x^{11}

\int 51963120 x^{9} d x = 5196312 x^{10}

\int 55427328 x^{8} d x = 6158592 x^{9}

\int 43110144 x^{7} d x = 5388768 x^{8}

\int 24634368 x^{6} d x = \frac{24634368 x ^{7}}{7}

\int 10264320 x^{5} d x = 1710720 x^{6}

\int 3041280 x^{4} d x = 608256 x^{5}

\int 608256 x^{3} d x = 152064 x^{4}

\int 73728 x^{2} d x = 24576 x^{3}

\int 4096 x d x = 2048 x^{2}

= \frac{531441 x ^{14}}{14} + \frac{4251528 x ^{13}}{13} + 1299078 x^{12} + 3149280 x^{11} + 5196312 x^{10} + 6158592 x^{9} + 5388768 x^{8} + \frac{24634368 x ^{7}}{7} + 1710720 x^{6} + 608256 x^{5} + 152064 x^{4} + 24576 x^{3} + 2048 x^{2}

Add a constant to the solution

= \frac{531441 x ^{14}}{14} + \frac{4251528 x ^{13}}{13} + 1299078 x^{12} + 3149280 x^{11} + 5196312 x^{10} + 6158592 x^{9} + 5388768 x^{8} + \frac{24634368 x ^{7}}{7} + 1710720 x^{6} + 608256 x^{5} + 152064 x^{4} + 24576 x^{3} + 2048 x^{2} + C

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Frequently Asked Questions (FAQ)

What is the integral of x(3x+2)^{12} ?
The integral of x(3x+2)^{12} is