Calculate the integral:

Question

(dx)/(x^(2)+5x+4)?

Answer 1

The integral expression is ʃ 1/(x² + 5x + 4) dx.

Write the quadratic expression in complete square form.

x² + 5x + 4

= (x² + 5x + 25/4) - 25/4 + 4

= (x + 5/2)²  - 9/4

= (x + 5/2)²  - (3/2)²

Let x + 5/2 = u ⇒ dx = du.

ʃ 1/(x² + 5x + 4) dx = ʃ 1/[u² - (3/2)²] du

Apply formula: ʃ 1/(u² - a²) du = (1/2a) ln|(u-a)/(u+a)| + C

ʃ 1/(x² + 5x + 4) dx = (1/3) ln|(u-3/2)/(u+3/2)| + C

ʃ 1/(x² + 5x + 4) dx = (1/3) ln|(x + 5/2-3/2)/(x + 5/2+3/2)| + C      [ Since x + 5/2 = u ]

ʃ 1/(x² + 5x + 4) dx = (1/3) ln|(x + 1)/(x + 4)| + C