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

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

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

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

ln x^{2} - 2 x + 2 + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

= ln x^{2} - 2 x + 2

Add a constant to the solution

= ln x^{2} - 2 x + 2 + C

\frac{d x}{d t} = 7 (x^{2} + 1), x (\frac{π}{4}) = 1

6 t^{2} - 4 t + 9

f (t) = 2 e^{- t} cos (4 t)

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

\frac{d}{d x} (12 x - x^{2})

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