How do you integrate #3/(x ^(2) + 16x +5)# using partial fractions?

Question

Emilia Aguilar · Accepted Answer

First factorize the denominator using the formula for the quadratic solution , which will drag some #sqrt(59)#s in. Are you sure the #5# shouldn't be a #15#?

Emma Aldridge · Answer

undefined To integrate $ \frac{3}{x^2 + 16x + 5} $ using partial fractions, you first factor the denominator $ x^2 + 16x + 5 $, then express $ \frac{3}{x^2 + 16x + 5} $ as a sum of partial fractions with denominators corresponding to the factors of the denominator. After that, solve for the unknown constants, and then integrate each partial fraction separately.

Here are the steps:

1. Factor the denominator: $ x^2 + 16x + 5 = (x + 1)(x + 5) $.

2. Express $ \frac{3}{x^2 + 16x + 5} $ as partial fractions: $ \frac{3}{x^2 + 16x + 5} = \frac{A}{x + 1} + \frac{B}{x + 5} $, where $ A $ and $ B $ are constants to be determined.

3. Multiply both sides by $ x^2 + 16x + 5 $ to clear the fractions: 
$$ 3 = A(x + 5) + B(x + 1) $$.

4. Expand and collect like terms:
$$ 3 = (A + B)x + (5A + B) $$.

5. Equate the coefficients of like terms:
$$ A + B = 0 $$ (for the $ x $ term),
$$ 5A + B = 3 $$ (for the constant term).

6. Solve the system of equations to find $ A $ and $ B $.

7. Once you have found the values of $ A $ and $ B $, integrate each partial fraction separately.

8. Finally, combine the integrals to obtain the result of the original integral.