Question Detail
Question
Evaluate ∫ dx / (x2 + 4x + 7)
Correct Answer
Option B is the correct answer.
Detailed Explanation
To evaluate the integral, we first complete the square in the denominator:x2 + 4x + 7 = (x2 + 4x + 4) + 3 = (x + 2)2 + 3So the integral becomes ∫ dx / ((x + 2)2 + 3).This integral is in the form ∫ du / (u2 + a2), which integrates to (1/a) tan-1(u/a) + C.Here, u = x + 2, so du = dx. And a2 = 3, so a = √3.Substitute these values into the formula:(1/√3) tan-1((x + 2)/√3) + CThis matches option B.
Hint
Complete the square for the quadratic expression in the denominator to transform the integral into the form ∫ du / (u2 + a2).