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

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

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

integral

\frac{2 - x}{1 - x}

x - ln ∣ - x + 1 ∣ - 1 + C

Solution steps

\int \frac{2 - x}{1 - x} d x

Expand

\frac{2 - x}{1 - x} : \frac{2}{1 - x} - \frac{x}{1 - 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{2}{1 - x} d x - \int \frac{x}{1 - x} d x

\int \frac{2}{1 - x} d x = - 2 ln ∣ 1 - x ∣

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

= - 2 ln ∣ 1 - x ∣ - (1 - x - ln ∣ 1 - x ∣)

Simplify

= x - ln ∣ - x + 1 ∣ - 1

Add a constant to the solution

= x - ln ∣ - x + 1 ∣ - 1 + C

\frac{tan ( x )}{( 1 + cos ^{4} ( x ) )}

2 e^{- 2 x}

\frac{1}{( x )}

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

\frac{1}{1 + ( 7 x + 6 ) ^{2}}

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