Math Questions and Answers

Start Your Free Trial

Evaluate the integral integrate of ((3)/(x^2-x-2))dx

Expert Answers info

lfryerda eNotes educator | Certified Educator

calendarEducator since 2012

write738 answers

starTop subjects are Math and Science

To integrate this, we need to separate the function into simpler functions using partial fractions, then integrate each function separately.

`int 3/{x^2-x-2}dx`  factor denominator

`=int 3/{(x-2)(x+1)}dx`   split into partial fractions

this means that `3/{(x-2)(x+1)}=A/{x-2}+B/{x+1}` .  Multiplying out and comparing on the constant terms and the linear terms, we see that:

`3=A(x+1)+B(x-2)`  which gives the two equations:

`3=A-2B`   equation (1)

`0=A+B`   equation (2)

Subtract (1) from (2)

`3B=3`  so B=1 and A=-1

Now the integral becomes

`-int 1/{x-2} dx + int1/{x+1}dx`  integrate each term

`=-ln(x-2)+ln(x+1)+C`  where C is the constant of integration

`=ln({x+1}/{x-2})+C`

The integral evaluates to `ln({x+1}/{x-2})+C` .

check Approved by eNotes Editorial