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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (2 x + y^{2})

f (x) = x^{3}, 0, 4

y^{^{'}} - 3 y = - 6 y^{3}

\int_{0}^{1} \frac{4}{1 - x ^{2}} d x

x \to 0 lim (\frac{x}{x ^{2} - 9 x})

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