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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add similar elements:

arctan (\frac{1}{2} (x - 1)) - \frac{3}{2} arctan (\frac{1}{2} (x - 1)) = - \frac{1}{2} arctan (\frac{1}{2} (x - 1))

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

Add a constant to the solution

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

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

\int \frac{5 x ^{2} - 8 x - 8}{5 x} d x

\int cosh (\frac{x}{3}) d x

\int tan (6 θ) d θ

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