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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

(3, 8), (- 7, 7)

\int sin (2 x) e^{c o s (2)} x d x

f (x) = x^{2} + 3 x - 3, at x = 2

\int_{0}^{1} e^{- 0.1 t} d t

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

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