what is the integral of (4x^2-2)/(x^2-6x-40)
First do a long division
(4x^2-2)/(x^2-6x-40) = 4 + (24x+158)/(x^2 - 6x - 40)
now separate (24x+158)/(x^2 - 6x - 40
into partial fractions so that
A/(x-10) + B(x+4) = (24x + 158)/((x-10)(x+4))
then
A(x+4) + B(x-10) = 24x + 158
let x = -4
-14B = 62
B = -31/7
let x = 10
14A = 398
A = 199/7
(4x^2-2)/(x^2-6x-40) = 4 + (199/7)/(x-10) - (31/7)/(x+4)
so the integral is
4x + (199/7)ln(x-10) - (31/7)ln(x+10) + C , where C is a constant
I have a typo, the last line should say:
4x + (199/7)ln(x-10) - (31/7)ln(x+4) + C , where C is a constant
To compute the integral of the given expression, you'll need to perform partial fraction decomposition first. Here's a step-by-step guide to help you find the integral:
Step 1: Factorize the denominator
The denominator of the given expression is x^2 - 6x - 40. To find its factors, we need to find two numbers that multiply to -40 (the last term) and add up to -6 (the coefficient of the x term). After some calculations, we can determine that the factors are (x - 10) and (x + 4). Therefore, we can rewrite the denominator as (x - 10)(x + 4).
Step 2: Express the fraction using partial fraction decomposition
Now that we know the factorization of the denominator, we can use partial fraction decomposition to express the fraction as a sum of simpler fractions with different denominators. We write the given expression as:
(4x^2 - 2) / ((x - 10)(x + 4))
Step 3: Write the partial fraction form
The partial fraction form for the above expression can be written as:
(4x+18)/(x-10) + (2x-14)/(x+4)
Step 4: Integrate each term separately
Now that we have rewritten the expression as simpler fractions, we can integrate each term separately. The integral of (4x+18)/(x-10) can be found by using the substitution method or integration by parts, depending on the method you prefer. Similarly, the integral of (2x-14)/(x+4) can also be found using either method.
By following these steps, you should now be able to find the integral of the given expression.