What is the integral of (5x-5)/(x^2+x-6) ?

The integral of (5x-5)/(x^2+x-6) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (5x-5)/(x^2+x-6)

\int \frac{5 x - 5}{x ^{2} + x - 6} d x

\frac{5}{2} ln 4 x^{2} + 4 x - 24 + \frac{1}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x + C

Solution steps

\int \frac{5 x - 5}{x ^{2} + x - 6} d x

Expand

\frac{5 x - 5}{x ^{2} + x - 6} : \frac{5 x}{x ^{2} + x - 6} - \frac{5}{x ^{2} + x - 6}

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{5 x}{x ^{2} + x - 6} d x - \int \frac{5}{x ^{2} + x - 6} d x

\int \frac{5 x}{x ^{2} + x - 6} d x = 10 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x))

\int \frac{5}{x ^{2} + x - 6} d x = - ln \frac{6}{5} + \frac{2}{5} x + ln - \frac{4}{5} + \frac{2}{5} x

= 10 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - (- ln \frac{6}{5} + \frac{2}{5} x + ln - \frac{4}{5} + \frac{2}{5} x)

Simplify

10 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - (- ln \frac{6}{5} + \frac{2}{5} x + ln - \frac{4}{5} + \frac{2}{5} x) : \frac{5}{2} ln 4 x^{2} + 4 x - 24 + \frac{1}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x

= \frac{5}{2} ln 4 x^{2} + 4 x - 24 + \frac{1}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x

Add a constant to the solution

= \frac{5}{2} ln 4 x^{2} + 4 x - 24 + \frac{1}{2} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x + C

\int \frac{1}{x + 5} d x

\int 144 - 16 x^{2} d x

\int x a r tan (x) d x

\int (3 cos (x) + 2 sin (x)) d x

\int \frac{4 x}{x - 5} d x