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Popular >

integral of (x+2)^{13}

Solution

\int (x + 2)^{13} d x

Solution

\frac{x ^{14}}{14} + 2 x^{13} + 26 x^{12} + 208 x^{11} + 1144 x^{10} + 4576 x^{9} + 13728 x^{8} + \frac{219648 x ^{7}}{7} + 54912 x^{6} + 73216 x^{5} + 73216 x^{4} + 53248 x^{3} + 26624 x^{2} + 8192 x + C

Solution steps

\int (x + 2)^{13} d x

Expand

(x + 2)^{13} : x^{13} + 26 x^{12} + 312 x^{11} + 2288 x^{10} + 11440 x^{9} + 41184 x^{8} + 109824 x^{7} + 219648 x^{6} + 329472 x^{5} + 366080 x^{4} + 292864 x^{3} + 159744 x^{2} + 53248 x + 8192

= \int x^{13} + 26 x^{12} + 312 x^{11} + 2288 x^{10} + 11440 x^{9} + 41184 x^{8} + 109824 x^{7} + 219648 x^{6} + 329472 x^{5} + 366080 x^{4} + 292864 x^{3} + 159744 x^{2} + 53248 x + 8192 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 x^{13} d x + \int 26 x^{12} d x + \int 312 x^{11} d x + \int 2288 x^{10} d x + \int 11440 x^{9} d x + \int 41184 x^{8} d x + \int 109824 x^{7} d x + \int 219648 x^{6} d x + \int 329472 x^{5} d x + \int 366080 x^{4} d x + \int 292864 x^{3} d x + \int 159744 x^{2} d x + \int 53248 x d x + \int 8192 d x

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

\int 26 x^{12} d x = 2 x^{13}

\int 312 x^{11} d x = 26 x^{12}

\int 2288 x^{10} d x = 208 x^{11}

\int 11440 x^{9} d x = 1144 x^{10}

\int 41184 x^{8} d x = 4576 x^{9}

\int 109824 x^{7} d x = 13728 x^{8}

\int 219648 x^{6} d x = \frac{219648 x ^{7}}{7}

\int 329472 x^{5} d x = 54912 x^{6}

\int 366080 x^{4} d x = 73216 x^{5}

\int 292864 x^{3} d x = 73216 x^{4}

\int 159744 x^{2} d x = 53248 x^{3}

\int 53248 x d x = 26624 x^{2}

\int 8192 d x = 8192 x

= \frac{x ^{14}}{14} + 2 x^{13} + 26 x^{12} + 208 x^{11} + 1144 x^{10} + 4576 x^{9} + 13728 x^{8} + \frac{219648 x ^{7}}{7} + 54912 x^{6} + 73216 x^{5} + 73216 x^{4} + 53248 x^{3} + 26624 x^{2} + 8192 x

Add a constant to the solution

= \frac{x ^{14}}{14} + 2 x^{13} + 26 x^{12} + 208 x^{11} + 1144 x^{10} + 4576 x^{9} + 13728 x^{8} + \frac{219648 x ^{7}}{7} + 54912 x^{6} + 73216 x^{5} + 73216 x^{4} + 53248 x^{3} + 26624 x^{2} + 8192 x + C

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Frequently Asked Questions (FAQ)

What is the integral of (x+2)^{13} ?
The integral of (x+2)^{13} is