What is the integral of (2x^2+x+3)/((x+1)(x^2+1)) ?

The integral of (2x^2+x+3)/((x+1)(x^2+1)) is 2ln|x+1|+arctan(x)+C

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integral of (2x^2+x+3)/((x+1)(x^2+1))

\int \frac{2 x ^{2} + x + 3}{( x + 1 ) ( x ^{2} + 1 )} d x

2 ln ∣ x + 1 ∣ + arctan (x) + C

Solution steps

\int \frac{2 x ^{2} + x + 3}{( x + 1 ) ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{2 x ^{2} + x + 3}{( x + 1 ) ( x ^{2} + 1 )} : \frac{2}{x + 1} + \frac{1}{x ^{2} + 1}

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

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

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

= 2 ln ∣ x + 1 ∣ + arctan (x)

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + arctan (x) + C

4^{- x^{8}}

\int 2 sin^{2} (x) d x

x^{3} + y^{2} x^{4} + 4 y^{^{'}} = 0

\int_{0}^{x} (e^{- t^{2}}) d t

x 1 + y^{2} d x = y 1 + x^{2} d y

What is the integral of (2x^2+x+3)/((x+1)(x^2+1)) ?
The integral of (2x^2+x+3)/((x+1)(x^2+1)) is 2ln|x+1|+arctan(x)+C