Evaluate the following indefinite integral

Question

Evaluate the following indefinite integral:

 

∫1/(x²-3x-10) dx

Answer 1

Since the denominator factors into (x - 5)(x + 2) use partial fractions.

[1/(x-5)(x + 2)] = A/(x-5) + B(x+2)

1 = A(x+2) +B(x-5)

At this point you could expand and simplify the right side then equate coefficients, but probably just as easy to...

Let x = -2:

1 = B(-7)

B = -1/7

Let x = 5:

1 = 7A

A = 1/7

So original can be rewritten as:

∫ [(1/7)/(x-5) + (-1/7)/(x+2)]dx

(1/7) ∫ [1/(x - 5) - 1/(x+2)]dx

= (1/7) [ln|x - 5| - ln|x+2| ] + C

or (1/7) ln| (x-5)/(x+2) | + C