GIKI Entry Test Mathematics: Integration MCQs

GIKI Entry Test Mathematics: Integration MCQs — sample questions

1. Question 1

   Q1. ∫(2x + 1) / (x² + x + 1) dx

   • A) ln|x² + x + 1| + C
   • B) ln|x + 1| + C
   • C) (1/2)ln|x² + x + 1| + C
   • D) (1/2)ln|x + 1| + C

   Answer: ln|x² + x + 1| + C

   Explanation: Using substitution u = x² + x + 1, du/dx = 2x + 1, hence ∫(2x + 1) / (x² + x + 1) dx = ∫du/u = ln|u| + C.

2. Question 2

   Q2. ∫(sin x + cos x) / √(sin 2x) dx

   • A) √(sin x + cos x) + C
   • B) √(sin 2x) + C
   • C) sin^-1(sin x - cos x) + C
   • D) ln|sin x + cos x| + C

   Answer: sin^-1(sin x - cos x) + C

   Explanation: Using trigonometric identity sin 2x = (sin x + cos x)² - 1, and substitution, we simplify the integral to ∫1 / √(1 - (sin x - cos x)²) d(sin x - cos x).

3. Question 3

   Q3. ∫1 / (x² - 4) dx on 1 / 2 to 1

   • A) (1/4)ln|3|
   • B) (1/4)ln|(3/5)|
   • C) (1/4)ln|(5/3)|
   • D) -(1/4)ln|(3/5)|

   Answer: (1/4)ln|(3/5)|

   Explanation: Using partial fractions, 1 / (x² - 4) = 1/4 * (1/(x-2) - 1/(x+2)), and then integrating within the given limits.

4. Question 4

   Q4. ∫x² / √(x² + 4) dx

   • A) (x/2)√(x² + 4) - 2ln|x + √(x² + 4)| + C
   • B) (x/2)√(x² + 4) + 2ln|x + √(x² + 4)| + C
   • C) (x/4)√(x² + 4) + 2ln|x + √(x² + 4)| + C
   • D) (x/4)√(x² + 4) - 2ln|x + √(x² + 4)| + C

   Answer: (x/2)√(x² + 4) - 2ln|x + √(x² + 4)| + C

   Explanation: Using trigonometric substitution x = 2tan(θ), dx = 2sec²(θ)dθ, and simplifying.

5. Question 5

   Q5. ∫e^x (1 + x) / (1 + x)² dx

   • A) e^x / (1 + x) + C
   • B) e^x(1 + x) + C
   • C) e^x + C
   • D) e^x(1 - x) + C

   Answer: e^x / (1 + x) + C

   Explanation: Simplifying the integrand to e^x / (1 + x), then using substitution u = 1 + x.