What is the integral of (x-1)/(x^2-3) ?

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

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integral of (x-1)/(x^2-3)

\int \frac{x - 1}{x ^{2} - 3} d x

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

Solution steps

\int \frac{x - 1}{x ^{2} - 3} d x

Expand

\frac{x - 1}{x ^{2} - 3} : \frac{x}{x ^{2} - 3} - \frac{1}{x ^{2} - 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} - 3} d x - \int \frac{1}{x ^{2} - 3} d x

\int \frac{x}{x ^{2} - 3} d x = \frac{1}{2} ln x^{2} - 3

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

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

Simplify

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

Add a constant to the solution

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

\frac{1}{5 x ^{2} + 3 x}

10 y^{^{''}} + 1000 y = 130 cos (8 t)

\frac{3}{( s - 3 ) ^{3}}

\int e^{2 x} (1 - 5 e^{2 x})^{2} d x

\frac{\partial}{\partial x} (e^{- 2 x^{2} - 3 y^{2}})

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