Integration of 5/(x^2 + 2x + 4)

To integrate the given function, we need to express it in a form that can be easily integrated. We can do this by completing the square in the denominator.

5/(x^2 + 2x + 4) = 5/[(x + 1)^2 + 3]

Now, we can make a substitution u = x + 1 and du/dx = 1 to simplify the integration.

5/[(x + 1)^2 + 3] dx = 5/(u^2 + 3) du

We can now integrate using the formula ∫du/(u^2 + a^2) = (1/a) arctan(u/a) + C.

∫ 5/[(x + 1)^2 + 3] dx = ∫ 5/(u^2 + 3) du
                       = (5/sqrt(3)) arctan(u/sqrt(3)) + C
                       = (5/sqrt(3)) arctan((x + 1)/sqrt(3)) + C

Therefore, the integral of 5/(x^2 + 2x + 4) is (5/sqrt(3)) arctan((x + 1)/sqrt(3)) + C, where C is the constant of integration.