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

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

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

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

ln ∣ x + 1 ∣ + arctan (x) + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= ln ∣ x + 1 ∣ + arctan (x)

Add a constant to the solution

= ln ∣ x + 1 ∣ + arctan (x) + C

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

\int \frac{1}{2 x ^{3}} - \frac{1}{x ^{4}} d x

\frac{d}{d x} (\frac{sec ( x )}{8 + sec ( x )})

f (x) = - csc (x) cot (x)

sin (x^{0.5})

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