What is the integral of ((x-5))/(x^2+2x-3) ?

The integral of ((x-5))/(x^2+2x-3) is

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integral of ((x-5))/(x^2+2x-3)

\int \frac{( x - 5 )}{x ^{2} + 2 x - 3} d x

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

Solution steps

\int \frac{x - 5}{x ^{2} + 2 x - 3} d x

Expand

\frac{x - 5}{x ^{2} + 2 x - 3} : \frac{x}{x ^{2} + 2 x - 3} - \frac{5}{x ^{2} + 2 x - 3}

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{x}{x ^{2} + 2 x - 3} d x - \int \frac{5}{x ^{2} + 2 x - 3} d x

\int \frac{x}{x ^{2} + 2 x - 3} d x = \frac{1}{2} ln x^{2} + 2 x - 3 + \frac{1}{4} (ln \frac{1}{2} (x + 1) + 1 - ln \frac{1}{2} (x + 1) - 1)

\int \frac{5}{x ^{2} + 2 x - 3} d x = - \frac{5}{4} (ln \frac{x + 1}{2} + 1 - ln \frac{x + 1}{2} - 1)

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 2 x - 3 + \frac{1}{4} (ln \frac{1}{2} (x + 1) + 1 - ln \frac{1}{2} (x + 1) - 1) + \frac{5}{4} (ln \frac{1}{2} (x + 1) + 1 - ln \frac{1}{2} (x + 1) - 1) + C

\frac{( e ^{x} + e ^{- x} )}{2}

\frac{d}{d x} (\frac{x - 2}{3})

\frac{d}{d x} (- sin^{3} (x))

\int_{0}^{1} 8 x y

\int_{0}^{4} π (y - \frac{y ^{2}}{4}) d y

What is the integral of ((x-5))/(x^2+2x-3) ?
The integral of ((x-5))/(x^2+2x-3) is