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

The integral of (2-x)/(x^2+2x-3) is 1/4 ln|x-1|-5/4 ln|x+3|+C

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

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

\frac{1}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{2 - x}{x ^{2} + 2 x - 3} : \frac{1}{4 ( x - 1 )} - \frac{5}{4 ( x + 3 )}

= \int \frac{1}{4 ( x - 1 )} - \frac{5}{4 ( 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}{4 ( x - 1 )} d x - \int \frac{5}{4 ( x + 3 )} d x

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{5}{4 ( x + 3 )} d x = \frac{5}{4} ln ∣ x + 3 ∣

= \frac{1}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ x - 1 ∣ - \frac{5}{4} ln ∣ x + 3 ∣ + C

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x d x + y e^{- x} d y = 0, y^{0} = 1

\int cos (5 x) cos (3 x) d x

\int x sin (\frac{nπ x}{2}) d x

What is the integral of (2-x)/(x^2+2x-3) ?
The integral of (2-x)/(x^2+2x-3) is 1/4 ln|x-1|-5/4 ln|x+3|+C