What is the integral of (x^2+5x+2)/((x+1)(x^2+1)) ?

The integral of (x^2+5x+2)/((x+1)(x^2+1)) is -ln|x+1|+ln|x^2+1|+3arctan(x)+C

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integral of (x^2+5x+2)/((x+1)(x^2+1))

\int \frac{x ^{2} + 5 x + 2}{( x + 1 ) ( x ^{2} + 1 )} d x

- ln ∣ x + 1 ∣ + ln x^{2} + 1 + 3 arctan (x) + C

Solution steps

\int \frac{x ^{2} + 5 x + 2}{( x + 1 ) ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{x ^{2} + 5 x + 2}{( x + 1 ) ( x ^{2} + 1 )} : - \frac{1}{x + 1} + \frac{2 x + 3}{x ^{2} + 1}

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

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

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

= - ln ∣ x + 1 ∣ + ln x^{2} + 1 + 3 arctan (x)

Add a constant to the solution

= - ln ∣ x + 1 ∣ + ln x^{2} + 1 + 3 arctan (x) + C

x \to - 9 lim (\frac{x ^{2} - 6}{9 - x})

ln (1 - 7 x)

\frac{d}{d x} (ln (1 + sin (2 x)))

\int \frac{x ^{3}}{x ^{3} - 7 x + 6} d x

\frac{\partial}{\partial y} (sin (3 x + 4 y))

What is the integral of (x^2+5x+2)/((x+1)(x^2+1)) ?
The integral of (x^2+5x+2)/((x+1)(x^2+1)) is -ln|x+1|+ln|x^2+1|+3arctan(x)+C