ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Integration (Set 2)

Take the ECAT Mathematics — Integration (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

1. Q1. ∫ dx/x³ equals:

   • A. −1/(2x²) + c ✓
   • B. −3/x⁴ + c
   • C. −1/(4x⁴) + c
   • D. 1/(2x²) + c

   Answer: A. −1/(2x²) + c

   Explanation: ∫ x⁻³ dx = x⁻²/(−2) + c = −1/(2x²) + c.

2. Q2. ∫ sin 4x dx equals:

   • A. cos 4x/4 + c
   • B. −cos 4x/4 + c ✓
   • C. −4 cos 4x + c
   • D. −cos 4x + c

   Answer: B. −cos 4x/4 + c

   Explanation: ∫ sin ax dx = −cos ax/a + c, so the answer is −cos 4x/4 + c.

3. Q3. ∫ e^(−3x) dx equals:

   • A. e^(−3x + 1)/(−3x + 1) + c
   • B. e^(−3x)/3 + c
   • C. −e^(−3x)/3 + c ✓
   • D. −3e^(−3x) + c

   Answer: C. −e^(−3x)/3 + c

   Explanation: ∫ e^(ax) dx = e^(ax)/a + c with a = −3.

4. Q4. ∫ cosec²x dx equals:

   • A. cot x + c
   • B. tan x + c
   • C. −cosec x + c
   • D. −cot x + c ✓

   Answer: D. −cot x + c

   Explanation: Since d/dx (cot x) = −cosec²x, ∫ cosec²x dx = −cot x + c.

5. Q5. ∫ sec x tan x dx equals:

   • A. tan x + c
   • B. sec x + c ✓
   • C. −sec x + c
   • D. sec²x/2 + c

   Answer: B. sec x + c

   Explanation: d/dx (sec x) = sec x tan x, so the integral is sec x + c.

The other 25 questions

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6. Q6. ∫ dx/(2x + 1) equals:

   • A. 2 ln|2x + 1| + c
   • B. ln|2x + 1| + c
   • C. −1/(2x + 1)² + c
   • D. ½ ln|2x + 1| + c

7. Q7. ∫ √x dx equals:

   • A. (2/3)x^(2/3) + c
   • B. (3/2)x^(3/2) + c
   • C. 1/(2√x) + c
   • D. (2/3)x^(3/2) + c

8. Q8. ∫ (x + 1)² dx equals:

   • A. (x + 1)³/3 + c
   • B. x³/3 + 1 + c
   • C. (x + 1)³ + c
   • D. 2(x + 1) + c

9. Q9. ∫ cos²x dx equals:

   • A. x + (sin 2x)/2 + c
   • B. x/2 − (sin 2x)/4 + c
   • C. x/2 + (sin 2x)/4 + c
   • D. (cos³x)/3 + c

10. Q10. ∫ x sin x dx equals:

    • A. x cos x − sin x + c
    • B. −x cos x − sin x + c
    • C. (x²/2) sin x + c
    • D. sin x − x cos x + c

11. Q11. ∫ x ln x dx equals:

    • A. (x²/2) ln x + x²/4 + c
    • B. x ln x − x + c
    • C. (x²/2) ln x − x²/2 + c
    • D. (x²/2) ln x − x²/4 + c

12. Q12. ∫ eˣ(1/x − 1/x²) dx equals:

    • A. eˣ/x² + c
    • B. eˣ ln x + c
    • C. eˣ/x + c
    • D. −eˣ/x + c

13. Q13. ∫ x·e^(x²) dx equals:

    • A. ½ e^(x²) + c
    • B. e^(x²) + c
    • C. (x²/2) e^(x²) + c
    • D. 2e^(x²) + c

14. Q14. ∫ cos x/(1 + sin x) dx equals:

    • A. ln|1 + sin x| + c
    • B. 1/(1 + sin x)² + c
    • C. ln|cos x| + c
    • D. −ln|1 + sin x| + c

15. Q15. ∫ sec²x/tan x dx equals:

    • A. −1/tan x + c
    • B. ln|sec x| + c
    • C. ln|tan x| + c
    • D. tan²x/2 + c

16. Q16. ∫ sin³x cos x dx equals:

    • A. sin⁴x/4 + c
    • B. cos⁴x/4 + c
    • C. sin²x/2 + c
    • D. −sin⁴x/4 + c

17. Q17. ∫ dx/(1 + 4x²) equals:

    • A. 2 tan⁻¹(2x) + c
    • B. ¼ tan⁻¹(x/2) + c
    • C. ½ tan⁻¹(2x) + c
    • D. tan⁻¹(2x) + c

18. Q18. ∫ dx/(x² + 2x + 2) equals:

    • A. tan⁻¹(x + 2) + c
    • B. ½ tan⁻¹((x + 1)/2) + c
    • C. ln(x² + 2x + 2) + c
    • D. tan⁻¹(x + 1) + c

19. Q19. ∫ dx/[(x − 1)(x − 2)] equals:

    • A. ½ ln|(x − 2)/(x − 1)| + c
    • B. ln|(x − 2)/(x − 1)| + c
    • C. ln|(x − 1)(x − 2)| + c
    • D. ln|(x − 1)/(x − 2)| + c

20. Q20. ∫ x/(x + 1) dx equals:

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

21. Q21. ∫ (ln x)/x dx equals:

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

22. Q22. ∫ sin⁻¹x dx equals:

    • A. x sin⁻¹x + x/√(1 − x²) + c
    • B. x sin⁻¹x − √(1 − x²) + c
    • C. 1/√(1 − x²) + c
    • D. x sin⁻¹x + √(1 − x²) + c

23. Q23. For x > 0, ∫ e^(ln x) dx equals:

    • A. 1/x + c
    • B. e^(ln x) + c
    • C. x ln x + c
    • D. x²/2 + c

24. Q24. ∫ dx/(1 + cos x) equals:

    • A. tan(x/2) + c
    • B. 2 tan(x/2) + c
    • C. ln|1 + cos x| + c
    • D. ½ tan(x/2) + c

25. Q25. ∫ (eˣ + e⁻ˣ) dx equals:

    • A. 2eˣ + c
    • B. eˣ − e⁻ˣ + c
    • C. e⁻ˣ − eˣ + c
    • D. eˣ + e⁻ˣ + c

26. Q26. ∫ sin 3x cos 2x dx equals:

    • A. (cos 5x)/10 + (cos x)/2 + c
    • B. −(cos 5x)/5 − cos x + c
    • C. −(sin 5x)/10 − (sin x)/2 + c
    • D. −(cos 5x)/10 − (cos x)/2 + c

27. Q27. ∫ tan²x dx equals:

    • A. tan³x/3 + c
    • B. tan x − x + c
    • C. sec²x − x + c
    • D. tan x + x + c

28. Q28. If f′(x) = 2x + 1 and f(1) = 5, then f(x) equals:

    • A. x² + 3
    • B. x² + x + 5
    • C. 2x² + x + 2
    • D. x² + x + 3

29. Q29. ∫ (x³ − 1)/(x − 1) dx, x ≠ 1, equals:

    • A. x³/3 + x² + x + c
    • B. x³/3 + x²/2 + x + c
    • C. x³/3 − x + c
    • D. x⁴/4 − x + c

30. Q30. ∫ sinh x dx equals:

    • A. sinh x + c
    • B. −cosh x + c
    • C. cosh x + c
    • D. −sinh x + c