Question Detail

First, complete the square in the denominator: x2 + 4x + 7 = (x2 + 4x + 4) + 3 = (x+2)2 + 3.The integral becomes ∫ dx / ((x+2)2 + 3).This is in the standard form for the integral of 1/(u2 + a2), which is (1/a)arctan(u/a) + C.Here, let u = x+2, so du = dx. And a2 = 3, so a = √3.Substituting these values, we get (1/√3)arctan((x+2)/√3) + C.Note: arctan is also denoted as tan-1.