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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

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

= 2 \cdot \int \frac{- x + 1}{2 ( x ^{2} + 1 )} + \frac{1}{2 ( x + 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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (e^{x^{2} + y^{2} + 2 y + 5})

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

\int (x - \frac{2}{x}) d x

x \to 0 lim (x^{2} - 8 x + 9)

e^{6}

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