Indefinite integral

Question

∫(1)/x²-3x-10 dx

Answer 1

Hi Joe,

 

Integrals of this form can be solved via partial fraction decomposition. Recall that the denominator can be factored into the following form:

 

x² - 3x - 10 = (x - 5)(x + 2)

 

Now we can create our partial fractions as 

 

1/(x2 - 3x - 10) = A/(x-5) + B/(x + 2)

 

If we multiply the denominator on the left side over to the right side, we get

 

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

1 = Ax + 2A + Bx -5B

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

 

Now, we compare coefficients. There is no x on the left side, which means that A + B = 0 so that it can also vanish on the right side. This tells us that A = -B

 

What remains is now that 1 = 2A - 5B, or 1 = 2(-B) - 5B = -7B

So, B = -1/7 and A = 1/7

 

Now we have ∫1/(x2 - 3x - 10)dx = (1/7)∫1/(x - 5)dx - (1/7)∫1/(x + 2)dx  where I simply pulled out the values of A and B from the integrals since they are constants.

 

Doing the integration, you get  (1/7)ln(x - 5) - (1/7)ln(x + 2) + C

 

Factoring out the 1/7 and using the log property that ln(a) - ln(b) = ln(a/b) you get

 

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

 

 

Hope this helps!

-Ulisses