ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Integration (Set 3)

Take the ECAT Mathematics — Integration (Set 3): 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. ∫ (3x² − 2x + 1) dx equals:

   • A. x³ − x² + C
   • B. 6x − 2 + C
   • C. x³ − x² + x + C ✓
   • D. x³ − 2x² + x + C

   Answer: C. x³ − x² + x + C

   Explanation: Integrate term by term: x³ − x² + x + C.

2. Q2. ∫ ∛x dx equals:

   • A. (3/4)x^(4/3) + C ✓
   • B. (4/3)x^(4/3) + C
   • C. (3/2)x^(2/3) + C
   • D. (1/3)x^(−2/3) + C

   Answer: A. (3/4)x^(4/3) + C

   Explanation: ∛x = x^(1/3); add 1 to the power to get 4/3 and divide by 4/3: (3/4)x^(4/3).

3. Q3. ∫ e^(5x + 2) dx equals:

   • A. 5e^(5x+2) + C
   • B. e^(5x+2) + C
   • C. e^(5x+2)/5 + C ✓
   • D. e^(5x+2)/(5x + 2) + C

   Answer: C. e^(5x+2)/5 + C

   Explanation: ∫ e^(ax+b) dx = e^(ax+b)/a, here a = 5.

4. Q4. For x > 2/3, ∫ dx/(3x − 2) equals:

   • A. (1/3) ln(3x − 2) + C ✓
   • B. ln(3x − 2) + C
   • C. 3 ln(3x − 2) + C
   • D. −1/(3x − 2)² + C

   Answer: A. (1/3) ln(3x − 2) + C

   Explanation: ∫ dx/(ax + b) = (1/a) ln(ax + b), so the answer is (1/3) ln(3x − 2).

5. Q5. ∫ cosh 2x dx equals:

   • A. sinh 2x + C
   • B. 2 sinh 2x + C
   • C. (1/2) sinh 2x + C ✓
   • D. (1/2) cosh 2x + C

   Answer: C. (1/2) sinh 2x + C

   Explanation: The derivative of sinh 2x is 2 cosh 2x, so divide by 2.

The other 25 questions

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6. Q6. ∫ x e^(−x²) dx equals:

   • A. (1/2)e^(−x²) + C
   • B. −e^(−x²) + C
   • C. −(1/2)e^(−x²) + C
   • D. −2e^(−x²) + C

7. Q7. ∫ x(x² + 3)⁴ dx equals:

   • A. x²(x² + 3)⁵/10 + C
   • B. (x² + 3)⁵/5 + C
   • C. (x² + 3)⁵/(5x) + C
   • D. (x² + 3)⁵/10 + C

8. Q8. ∫ sin x/(1 + cos x) dx equals:

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

9. Q9. ∫ sec²x/(1 + tan x) dx equals:

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

10. Q10. ∫ tan²x sec²x dx equals:

    • A. tan³x + C
    • B. sec³x/3 + C
    • C. tan x sec x + C
    • D. tan³x/3 + C

11. Q11. ∫ cos³x dx equals:

    • A. sin x − (sin³x)/3 + C
    • B. sin x + (sin³x)/3 + C
    • C. (cos⁴x)/4 + C
    • D. (sin⁴x)/4 + C

12. Q12. For −1 < x < 1, ∫ x/√(1 − x²) dx equals:

    • A. −√(1 − x²) + C
    • B. sin⁻¹x + C
    • C. −2√(1 − x²) + C
    • D. √(1 − x²) + C

13. Q13. ∫ (x² − 1)/(x² + 1) dx equals:

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

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

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

15. Q15. ∫ (2x + 1)/(x² + x + 5) dx equals:

    • A. tan⁻¹(x² + x + 5) + C
    • B. ln(x² + x + 5) + C
    • C. (1/2) ln(x² + x + 5) + C
    • D. 2 ln(x² + x + 5) + C

16. Q16. ∫ dx/(x² − 6x + 13) equals:

    • A. (1/2) tan⁻¹((x − 3)/2) + C
    • B. (1/3) tan⁻¹(x/3) + C
    • C. (1/2) tan⁻¹(x − 3) + C
    • D. tan⁻¹((x − 3)/2) + C

17. Q17. ∫ dx/√(9 − 4x²) equals:

    • A. (1/3) sin⁻¹(2x/3) + C
    • B. sin⁻¹(2x/3) + C
    • C. (1/2) sin⁻¹(2x/3) + C
    • D. (1/2) sin⁻¹(x/3) + C

18. Q18. ∫ dx/√(x² + 9) equals:

    • A. ½ ln(x² + 9) + C
    • B. (1/3) tan⁻¹(x/3) + C
    • C. sin⁻¹(x/3) + C
    • D. ln(x + √(x² + 9)) + C

19. Q19. For x > 1, ∫ (x + 3)/[(x − 1)(x + 1)] dx equals:

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

20. Q20. For x > 0, ∫ dx/[x(x² + 1)] equals:

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

21. Q21. ∫ x² cos x dx equals:

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

22. Q22. ∫ x sec²x dx equals:

    • A. (x²/2) tan x + C
    • B. x tan x + ln|cos x| + C
    • C. x tan x − ln|cos x| + C
    • D. x tan x − sec²x + C

23. Q23. ∫ x² ln x dx (x > 0) equals:

    • A. (x³/3) ln x − x³/9 + C
    • B. (x³/3) ln x + x³/9 + C
    • C. x³ ln x − x³/3 + C
    • D. (x³/3) ln x − x³/3 + C

24. Q24. For −1 ≤ x ≤ 1, ∫ cos⁻¹x dx equals:

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

25. Q25. ∫ eˣ(x² + 2x) dx equals:

    • A. (x² + 2)eˣ + C
    • B. x²eˣ + C
    • C. (x² + 2x)eˣ + C
    • D. (x² − 2x)eˣ + C

26. Q26. ∫ dx/(eˣ + e⁻ˣ) equals:

    • A. ln(eˣ + e⁻ˣ) + C
    • B. tan⁻¹(eˣ) + C
    • C. tan⁻¹(e⁻ˣ) + C
    • D. ½ ln(eˣ + e⁻ˣ) + C

27. Q27. For x > 0, ∫ (ln x)/x² dx equals:

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

28. Q28. ∫ e^(2x) cos x dx equals:

    • A. e^(2x)(2 cos x − sin x)/5 + C
    • B. e^(2x)(2 cos x + sin x)/3 + C
    • C. e^(2x)(cos x + 2 sin x)/5 + C
    • D. e^(2x)(2 cos x + sin x)/5 + C

29. Q29. ∫ sin 3x sin x dx equals:

    • A. −cos 3x cos x + C
    • B. sin 2x/2 − sin 4x/4 + C
    • C. sin 2x/4 − sin 4x/8 + C
    • D. sin 2x/4 + sin 4x/8 + C

30. Q30. If ∫ f(x) dx = x eˣ + C, then f(x) equals:

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