What is the integral of (x^2-14)^9(2x) ?

The integral of (x^2-14)^9(2x) is

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integral of (x^2-14)^9(2x)

\int (x^{2} - 14)^{9} (2 x) d x

2 (\frac{x ^{20}}{20} - 7 x^{18} + 441 x^{16} - 16464 x^{14} + 403368 x^{12} - \frac{33882912 x ^{10}}{5} + 79060128 x^{8} - 632481024 x^{6} + 3320525376 x^{4} - 10330523392 x^{2}) + C

Solution steps

\int (x^{2} - 14)^{9} \cdot 2 x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int (x^{2} - 14)^{9} x d x

Expand

(x^{2} - 14)^{9} x : x^{19} - 126 x^{17} + 7056 x^{15} - 230496 x^{13} + 4840416 x^{11} - 67765824 x^{9} + 632481024 x^{7} - 3794886144 x^{5} + 1 4^{8} \cdot 9 x^{3} - 1 4^{9} x

= 2 \cdot \int x^{19} - 126 x^{17} + 7056 x^{15} - 230496 x^{13} + 4840416 x^{11} - 67765824 x^{9} + 632481024 x^{7} - 3794886144 x^{5} + 1 4^{8} \cdot 9 x^{3} - 1 4^{9} 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

= 2 (\int x^{19} d x - \int 126 x^{17} d x + \int 7056 x^{15} d x - \int 230496 x^{13} d x + \int 4840416 x^{11} d x - \int 67765824 x^{9} d x + \int 632481024 x^{7} d x - \int 3794886144 x^{5} d x + \int 1 4^{8} \cdot 9 x^{3} d x - \int 1 4^{9} x d x)

\int x^{19} d x = \frac{x ^{20}}{20}

\int 126 x^{17} d x = 7 x^{18}

\int 7056 x^{15} d x = 441 x^{16}

\int 230496 x^{13} d x = 16464 x^{14}

\int 4840416 x^{11} d x = 403368 x^{12}

\int 67765824 x^{9} d x = \frac{33882912 x ^{10}}{5}

\int 632481024 x^{7} d x = 79060128 x^{8}

\int 3794886144 x^{5} d x = 632481024 x^{6}

\int 1 4^{8} \cdot 9 x^{3} d x = 3320525376 x^{4}

\int 1 4^{9} x d x = 10330523392 x^{2}

= 2 (\frac{x ^{20}}{20} - 7 x^{18} + 441 x^{16} - 16464 x^{14} + 403368 x^{12} - \frac{33882912 x ^{10}}{5} + 79060128 x^{8} - 632481024 x^{6} + 3320525376 x^{4} - 10330523392 x^{2})

Add a constant to the solution

= 2 (\frac{x ^{20}}{20} - 7 x^{18} + 441 x^{16} - 16464 x^{14} + 403368 x^{12} - \frac{33882912 x ^{10}}{5} + 79060128 x^{8} - 632481024 x^{6} + 3320525376 x^{4} - 10330523392 x^{2}) + C

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