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

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

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

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

arctan (x) + \frac{1}{x + 1} + C

Solution steps

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

Take the constant out:

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

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

Take the partial fraction of

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

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

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

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

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

Simplify

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

= arctan (x) + \frac{1}{x + 1}

Add a constant to the solution

= arctan (x) + \frac{1}{x + 1} + C

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