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MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate ∫ 1 / (x2 + 4x + 7) dx
Options
A (1/√2)tan<sup>-1</sup>((x+3)/√2) + C
B (1/√3)tan<sup>-1</sup>((x+2)/√3) + C
Correct Answer
C (1/√6)tan<sup>-1</sup>((x+7)/√6) + C
D (1/√5)tan<sup>-1</sup>((x+6)/√5) + C
Correct Answer

Option B is the correct answer.

Detailed Explanation

This integral involves a quadratic in the denominator. We should complete the square.The denominator is x2 + 4x + 7.To complete the square for x2 + 4x, we add and subtract (4/2)2 = 22 = 4:x2 + 4x + 4 - 4 + 7 = (x+2)2 + 3So the integral becomes: ∫ 1 / ((x+2)2 + (√3)2) dxThis is in the form of the standard integral ∫ 1 / (u2 + a2) du = (1/a) tan-1(u/a) + C.Here, u = x+2 (so du = dx) and a = √3.Substitute these values:= (1/√3) tan-1((x+2)/√3) + CThis matches option B.

Hint

Complete the square in the denominator to transform the integral into a standard form involving inverse tangent.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 770 of 930
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