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MTH 102 MTH 102 Past Questions Objective Question
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Question

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

Option B is the correct answer.

Detailed Explanation

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 integral is of the standard form ∫ du / (u2 + a2) = (1⁄a) tan-1(u⁄a) + C.Here, u = x+2 (so du = dx) and a2 = 3, which means a = √3.Substituting these values:= 1⁄√3 tan-1((x+2)⁄√3) + C.This matches option B.

Hint

Complete the square in the denominator to transform the integral into a standard arctan form. Remember ∫ du / (u2 + a2) = (1/a) arctan(u/a) + C.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 662 of 930
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