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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>⁄<sub>√2</sub> tan<sup>-1</sup>(<sup>x+3</sup>⁄<sub>√2</sub>) + C
B <sup>1</sup>⁄<sub>√3</sub> tan<sup>-1</sup>(<sup>x+2</sup>⁄<sub>√3</sub>) + C
Correct Answer
C <sup>1</sup>⁄<sub>√6</sub> tan<sup>-1</sup>(<sup>x+7</sup>⁄<sub>√6</sub>) + C
D <sup>1</sup>⁄<sub>√5</sub> tan<sup>-1</sup>(<sup>x+6</sup>⁄<sub>√5</sub>) + C
Correct Answer

Option B is the correct answer.

Detailed Explanation

To evaluate this integral, we first complete the square in the denominator. x2 + 4x + 7 = (x2 + 4x + 4) + 3 = (x+2)2 + 3. Now the integral becomes ∫ dx ⁄ ((x+2)2 + (√3)2). This is in the standard form for the integral of 1⁄(u2 + a2), which is (1⁄a)tan-1(u⁄a) + C. Here, u = x+2 (so du = dx) and a = √3. Substituting these values, we get (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 arctangent form.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 600 of 930
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