What is the integral of (x^2+5)/(x^3-x^2+x+3) ?

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

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integral of (x^2+5)/(x^3-x^2+x+3)

\int \frac{x ^{2} + 5}{x ^{3} - x ^{2} + x + 3} d x

ln ∣ x + 1 ∣ + 2 arctan (\frac{1}{2} (x - 1)) + C

Solution steps

\int \frac{x ^{2} + 5}{x ^{3} - x ^{2} + x + 3} d x

Take the partial fraction of

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

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

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

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

= ln ∣ x + 1 ∣ + 2 arctan (\frac{1}{2} (x - 1))

Add a constant to the solution

= ln ∣ x + 1 ∣ + 2 arctan (\frac{1}{2} (x - 1)) + C

\frac{d}{d x} (\frac{e ^{x}}{9 e ^{x} + x ^{2}})

\int_{2}^{3} 3 d x

- x^{2} + 10 x - 16, 2 x - 4, 2, 8

- 5 x^{3} ln (x)

\int - e^{- x} \frac{1}{x} d x

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