Find : ∫dx/(5 - 8x - x²)

To solve the integral ∫dx/(5 - 8x - x²), we first complete the square for the denominator:

5 - 8x - x² = -(x² + 8x - 5) = -(x² + 8x + 16 - 16 - 5) = -( (x + 4)² - 21 ) = 21 - (x + 4)²

Now, the integral becomes:

∫dx/(21 - (x + 4)²)

Let u = x + 4, then du = dx. The integral transforms to:

∫du/(21 - u²)

This is of the form ∫du/(a² - u²), where a² = 21, so a = √21. The solution to this integral is:

(1/(2a))ln|(a + u)/(a - u)| + C

Substituting back a = √21 and u = x + 4, we get:

(1/(2√21))ln| (√21 + x + 4)/(√21 - x - 4) | + C

Therefore, the solution to the integral is:

(1/(2√21))ln| (√21 + x + 4)/(√21 - x - 4) | + C