What is the integral from 3 to 6 of x/(x^2+x-2) ?

The integral from 3 to 6 of x/(x^2+x-2) is 2ln(2)+(-ln(5)-ln(2))/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 3 to 6 of x/(x^2+x-2)

\int_{3}^{6} \frac{x}{x ^{2} + x - 2} d x

2 ln (2) + \frac{- ln ( 5 ) - ln ( 2 )}{3}

Decimal Notation

0.61876 \dots

Solution steps

\int_{3}^{6} \frac{x}{x ^{2} + x - 2} d x

Take the partial fraction of

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

= \int_{3}^{6} \frac{1}{3 ( x - 1 )} + \frac{2}{3 ( x + 2 )} 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_{3}^{6} \frac{1}{3 ( x - 1 )} d x + \int_{3}^{6} \frac{2}{3 ( x + 2 )} d x

\int_{3}^{6} \frac{1}{3 ( x - 1 )} d x = \frac{1}{3} (ln (5) - ln (2))

\int_{3}^{6} \frac{2}{3 ( x + 2 )} d x = 2 ln (2) - \frac{2}{3} ln (5)

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

Simplify

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

= 2 ln (2) + \frac{- ln ( 5 ) - ln ( 2 )}{3}

\int_{0}^{\frac{3}{4}} \frac{6}{x ( ln ( x ) ) ^{3}} d x

\int_{- 4 π}^{- \frac{7 π}{2}} sec (x) d x

\int_{1}^{5} 5 x^{2} - \frac{x}{3} d x

\int_{2}^{5} \frac{1}{x - 7} d x

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

What is the integral from 3 to 6 of x/(x^2+x-2) ?
The integral from 3 to 6 of x/(x^2+x-2) is 2ln(2)+(-ln(5)-ln(2))/3