ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Integration (Set 1)

Take the ECAT Mathematics — Integration (Set 1): 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. ∫ x⁴ dx equals:

   • A. 4x³ + c
   • B. x⁴/4 + c
   • C. x⁵ + c
   • D. x⁵/5 + c ✓

   Answer: D. x⁵/5 + c

   Explanation: Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c, so ∫ x⁴ dx = x⁵/5 + c.

2. Q2. ∫ cos 3x dx equals:

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

   Answer: A. (1/3) sin 3x + c

   Explanation: ∫ cos(ax) dx = sin(ax)/a + c with a = 3.

3. Q3. ∫ e^(2x) dx equals:

   • A. e^(2x) + c
   • B. e^(2x+1)/(2x + 1) + c
   • C. 2e^(2x) + c
   • D. ½ e^(2x) + c ✓

   Answer: D. ½ e^(2x) + c

   Explanation: ∫ e^(ax) dx = e^(ax)/a + c, so the answer is ½ e^(2x) + c.

4. Q4. ∫ sec²x dx equals:

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

   Answer: A. tan x + c

   Explanation: Since d/dx tan x = sec²x, ∫ sec²x dx = tan x + c.

5. Q5. ∫ dx/(1 + x²) equals:

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

   Answer: D. tan⁻¹x + c

   Explanation: d/dx tan⁻¹x = 1/(1 + x²), so the integral is tan⁻¹x + c.

The other 25 questions

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6. Q6. ∫ 2x/(x² + 1) dx equals:

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

7. Q7. ∫ x·eˣ dx equals:

   • A. x²eˣ/2 + c
   • B. eˣ(x − 1) + c
   • C. eˣ(x + 1) + c
   • D. x·eˣ + c

8. Q8. ∫ ln x dx equals:

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

9. Q9. ∫ x cos x dx equals:

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

10. Q10. ∫ tan x dx equals:

    • A. −ln|sec x| + c
    • B. ln|sec x| + c
    • C. sec²x + c
    • D. ln|sin x| + c

11. Q11. ∫ sin²x dx equals:

    • A. −cos²x + c
    • B. (sin³x)/3 + c
    • C. x/2 − (sin 2x)/4 + c
    • D. x/2 + (sin 2x)/4 + c

12. Q12. ∫ dx/(x² − 1) equals:

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

13. Q13. ∫ dx/[x(x + 1)] equals:

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

14. Q14. ∫ (2x + 3)⁵ dx equals:

    • A. 10(2x + 3)⁴ + c
    • B. (2x + 3)⁶/2 + c
    • C. (2x + 3)⁶/6 + c
    • D. (2x + 3)⁶/12 + c

15. Q15. ∫ x√(x² + 1) dx equals:

    • A. (2/3)(x² + 1)^(3/2) + c
    • B. (x² + 1)^(3/2) + c
    • C. (1/2)(x² + 1)^(1/2) + c
    • D. (1/3)(x² + 1)^(3/2) + c

16. Q16. ∫ eˣ(sin x + cos x) dx equals:

    • A. eˣ sin x + c
    • B. −eˣ sin x + c
    • C. eˣ(cos x − sin x) + c
    • D. eˣ cos x + c

17. Q17. ∫ dx/(x² + 9) equals:

    • A. (1/3) tan⁻¹(x/3) + c
    • B. tan⁻¹(x/3) + c
    • C. (1/9) tan⁻¹(x/9) + c
    • D. 3 tan⁻¹(3x) + c

18. Q18. ∫ dx/√(4 − x²) equals:

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

19. Q19. ∫ sin x cos x dx equals:

    • A. ½ sin²x + c
    • B. −½ sin²x + c
    • C. sin²x + c
    • D. ½ cos²x + c

20. Q20. ∫ x²eˣ dx equals:

    • A. eˣ(x² − 2x) + c
    • B. x³eˣ/3 + c
    • C. eˣ(x² − 2x + 2) + c
    • D. eˣ(x² + 2x + 2) + c

21. Q21. ∫ (3x + 5)/[(x + 1)(x + 2)] dx equals:

    • A. 2 ln|x + 1| − ln|x + 2| + c
    • B. 3 ln|x + 1| + 5 ln|x + 2| + c
    • C. 2 ln|x + 1| + ln|x + 2| + c
    • D. ln|x + 1| + 2 ln|x + 2| + c

22. Q22. ∫ 3ˣ dx equals:

    • A. 3ˣ/ln 3 + c
    • B. x·3ˣ⁻¹ + c
    • C. 3ˣ⁺¹/(x + 1) + c
    • D. 3ˣ ln 3 + c

23. Q23. ∫ (x² + 1)/x dx equals:

    • A. x²/2 − 1/x² + c
    • B. (x³/3 + x) ln|x| + c
    • C. x + ln|x| + c
    • D. x²/2 + ln|x| + c

24. Q24. ∫ eˣ/(1 + eˣ) dx equals:

    • A. eˣ ln(1 + eˣ) + c
    • B. x − eˣ + c
    • C. ln(1 + eˣ) + c
    • D. 1/(1 + eˣ) + c

25. Q25. ∫ dx/(x ln x) equals:

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

26. Q26. ∫ tan⁻¹x dx equals:

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

27. Q27. If ∫ f(x) dx = x³ + sin x + c, then f(x) equals:

    • A. 3x² + cos x
    • B. 3x² − cos x
    • C. x⁴/4 − cos x
    • D. x² + cos x

28. Q28. ∫ sec x dx equals:

    • A. ln|sec x + tan x| + c
    • B. sec x tan x + c
    • C. ln|sec x| + c
    • D. tan x + c

29. Q29. ∫ cot x dx equals:

    • A. −ln|sin x| + c
    • B. −cosec²x + c
    • C. ln|sin x| + c
    • D. ln|cos x| + c

30. Q30. ∫ dx/(x² − 4) equals:

    • A. (1/4) ln|(x + 2)/(x − 2)| + c
    • B. (1/4) ln|(x − 2)/(x + 2)| + c
    • C. (1/2) ln|(x − 2)/(x + 2)| + c
    • D. (1/2) tan⁻¹(x/2) + c