What is the integral of x+x^{12}+x^{-12} ?

The integral of x+x^{12}+x^{-12} is (x^2)/2+(x^{13})/(13)-1/(11x^{11)}+C

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integral of x+x^{12}+x^{-12}

\int x + x^{12} + x^{- 12} d x

\frac{x ^{2}}{2} + \frac{x ^{13}}{13} - \frac{1}{11 x ^{11}} + C

Solution steps

\int x + x^{12} + 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 x d x + \int x^{12} d x + \int x^{- 12} d x

\int x d x = \frac{x ^{2}}{2}

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

\int x^{- 12} d x = - \frac{1}{11 x ^{11}}

= \frac{x ^{2}}{2} + \frac{x ^{13}}{13} - \frac{1}{11 x ^{11}}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{x ^{13}}{13} - \frac{1}{11 x ^{11}} + C

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What is the integral of x+x^{12}+x^{-12} ?
The integral of x+x^{12}+x^{-12} is (x^2)/2+(x^{13})/(13)-1/(11x^{11)}+C