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{d}{d x} (\frac{2 x}{- 8 x - 1})

y^{^{'}} - \frac{2}{t} y = t^{2} e^{t}

cos^{2} (x) sin (x) (\frac{d y}{d x}) + cos^{3} (x) y = 1

\frac{d}{d y} (x^{6 y})

x \to - 3 lim (\frac{x + 3}{x - 3})

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