What is the integral of (x^6+x+1+x^{-1}+x^{-2}) ?

The integral of (x^6+x+1+x^{-1}+x^{-2}) is (x^7)/7+(x^2)/2+x+ln|x|-1/x+C

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integral of (x^6+x+1+x^{-1}+x^{-2})

\int (x^{6} + x + 1 + x^{- 1} + x^{- 2}) d x

\frac{x ^{7}}{7} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x} + C

Solution steps

\int x^{6} + x + 1 + x^{- 1} + x^{- 2} d x

Simplify

x^{6} + x + 1 + x^{- 1} + x^{- 2} : x^{6} + x + 1 + \frac{1}{x} + \frac{1}{x ^{2}}

= \int x^{6} + x + 1 + \frac{1}{x} + \frac{1}{x ^{2}} 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^{6} d x + \int x d x + \int 1 d x + \int \frac{1}{x} d x + \int \frac{1}{x ^{2}} d x

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

= \frac{x ^{7}}{7} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x} + C

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What is the integral of (x^6+x+1+x^{-1}+x^{-2}) ?
The integral of (x^6+x+1+x^{-1}+x^{-2}) is (x^7)/7+(x^2)/2+x+ln|x|-1/x+C