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Popular >

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

Solution

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

Solution

531441 x - 2125764 x^{2} + 5196312 x^{3} - 8660520 x^{4} + 10392624 x^{5} - 9237888 x^{6} + 6158592 x^{7} - 3079296 x^{8} + 1140480 x^{9} - 304128 x^{10} + 55296 x^{11} - 6144 x^{12} + \frac{4096 x ^{13}}{13} + C

Solution steps

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

Expand

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

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

\int 531441 d x = 531441 x

\int 4251528 x d x = 2125764 x^{2}

\int 15588936 x^{2} d x = 5196312 x^{3}

\int 34642080 x^{3} d x = 8660520 x^{4}

\int 51963120 x^{4} d x = 10392624 x^{5}

\int 55427328 x^{5} d x = 9237888 x^{6}

\int 43110144 x^{6} d x = 6158592 x^{7}

\int 24634368 x^{7} d x = 3079296 x^{8}

\int 10264320 x^{8} d x = 1140480 x^{9}

\int 3041280 x^{9} d x = 304128 x^{10}

\int 608256 x^{10} d x = 55296 x^{11}

\int 73728 x^{11} d x = 6144 x^{12}

\int 4096 x^{12} d x = \frac{4096 x ^{13}}{13}

= 531441 x - 2125764 x^{2} + 5196312 x^{3} - 8660520 x^{4} + 10392624 x^{5} - 9237888 x^{6} + 6158592 x^{7} - 3079296 x^{8} + 1140480 x^{9} - 304128 x^{10} + 55296 x^{11} - 6144 x^{12} + \frac{4096 x ^{13}}{13}

Add a constant to the solution

= 531441 x - 2125764 x^{2} + 5196312 x^{3} - 8660520 x^{4} + 10392624 x^{5} - 9237888 x^{6} + 6158592 x^{7} - 3079296 x^{8} + 1140480 x^{9} - 304128 x^{10} + 55296 x^{11} - 6144 x^{12} + \frac{4096 x ^{13}}{13} + C

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Frequently Asked Questions (FAQ)

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