What is the integral of (-x-2)/(x^2+1) ?

The integral of (-x-2)/(x^2+1) is -1/2 ln|x^2+1|-2arctan(x)+C

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integral of (-x-2)/(x^2+1)

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

- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x) + C

Solution steps

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{2}{x ^{2} + 1} d x = 2 arctan (x)

= - \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)

Add a constant to the solution

= - \frac{1}{2} ln x^{2} + 1 - 2 arctan (x) + C

\int (x - 2) e^{- x} d x

\int x sin (x + 2) d x

\int (sin (x) + e^{x}) d x

\int \frac{x sin ( x )}{cos ( x )} d x

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

What is the integral of (-x-2)/(x^2+1) ?
The integral of (-x-2)/(x^2+1) is -1/2 ln|x^2+1|-2arctan(x)+C