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

The integral of (cos(x)+x^{-1}-2x-1) is sin(x)+ln|x|-x^2-x+C

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

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

sin (x) + ln ∣ x ∣ - x^{2} - x + C

Solution steps

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

\int cos (x) d x = sin (x)

\int x^{- 1} d x = ln ∣ x ∣

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

\int 1 d x = x

= sin (x) + ln ∣ x ∣ - x^{2} - x

Add a constant to the solution

= sin (x) + ln ∣ x ∣ - x^{2} - x + C

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