ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Integration (Set 4)

Take the ECAT Mathematics — Integration (Set 4): 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 + 5) dx equals

   • A. 3x³ − 2x² + 5x + C
   • B. x³ − x² + 5x + C ✓
   • C. 6x − 2 + C
   • D. x³ − x² + 5 + C

   Answer: B. x³ − x² + 5x + C

   Explanation: Integrate term by term: 3x³/3 − 2x²/2 + 5x = x³ − x² + 5x.

2. Q2. ∫ sec²(3x) dx equals

   • A. 3 tan 3x + C
   • B. tan 3x + C
   • C. (1/3) sec 3x + C
   • D. (1/3) tan 3x + C ✓

   Answer: D. (1/3) tan 3x + C

   Explanation: d/dx tan 3x = 3 sec²3x, so divide by 3: (1/3) tan 3x + C.

3. Q3. ∫ e^(−2x) dx equals

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

   Answer: D. −e^(−2x)/2 + C

   Explanation: d/dx e^(−2x) = −2e^(−2x), so the antiderivative is −e^(−2x)/2.

4. Q4. ∫ dx/(2x + 5) equals

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

   Answer: A. (1/2) ln|2x + 5| + C

   Explanation: ∫ dx/(ax + b) = (1/a) ln|ax + b|, here a = 2.

5. Q5. ∫ cos 5x dx equals

   • A. 5 sin 5x + C
   • B. (sin 5x)/5 + C ✓
   • C. sin 5x + C
   • D. −(sin 5x)/5 + C

   Answer: B. (sin 5x)/5 + C

   Explanation: d/dx sin 5x = 5 cos 5x, so ∫ cos 5x dx = (sin 5x)/5.

The other 25 questions

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6. Q6. ∫ x e^(x²) dx equals

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

7. Q7. ∫ x sin 2x dx equals

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

8. Q8. ∫ (ln x)² dx equals

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

9. Q9. ∫ cot 2x dx equals

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

10. Q10. ∫ dx/(x² − 9) equals

    • A. (1/3) ln|(x − 3)/(x + 3)| + C
    • B. (1/6) ln|(x − 3)/(x + 3)| + C
    • C. (1/6) ln|(x + 3)/(x − 3)| + C
    • D. (1/6) ln|x² − 9| + C

11. Q11. ∫ dx/√(5 + 4x − x²) equals

    • A. cos⁻¹((x − 2)/3) + C
    • B. sin⁻¹(x/3) + C
    • C. sin⁻¹((x − 2)/5) + C
    • D. sin⁻¹((x − 2)/3) + C

12. Q12. ∫ dx/(x² + 4x + 13) equals

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

13. Q13. ∫ sin²x cos³x dx equals

    • A. cos⁴x/4 − cos⁶x/6 + C
    • B. sin³x/3 − sin⁵x/5 + C
    • C. sin³x/3 − cos³x/3 + C
    • D. sin³x/3 + sin⁵x/5 + C

14. Q14. ∫ dx/[x(1 + (ln x)²)] equals

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

15. Q15. ∫ sin³x dx equals

    • A. −cos x − (cos³x)/3 + C
    • B. (sin⁴x)/4 + C
    • C. −cos x + (cos³x)/3 + C
    • D. cos x − (cos³x)/3 + C

16. Q16. ∫ x/(x² + 1) dx equals

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

17. Q17. ∫ eˣ sin x dx equals

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

18. Q18. ∫ dx/(1 + sin x) equals

    • A. tan x − cos x + C
    • B. −cot(x/2) + C
    • C. tan x + sec x + C
    • D. tan x − sec x + C

19. Q19. ∫ x² eˣ dx equals

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

20. Q20. ∫ (x + 1)/((x − 1)(x + 2)) dx equals

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

21. Q21. ∫ x tan⁻¹x dx equals

    • A. (x²/2) tan⁻¹x − x/2 + C
    • B. ((x² − 1)/2) tan⁻¹x + x/2 + C
    • C. (x²/2) tan⁻¹x + ln(1 + x²)/4 + C
    • D. ((x² + 1)/2) tan⁻¹x − x/2 + C

22. Q22. ∫ dx/(x(x² + 1)) equals

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

23. Q23. ∫ eˣ(ln x + 1/x) dx equals

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

24. Q24. ∫ eˣ(tan x + sec²x) dx equals

    • A. eˣ(tan x − sec²x) + C
    • B. eˣ sec²x + C
    • C. eˣ tan x + C
    • D. eˣ sec x + C

25. Q25. ∫ √(9 − x²) dx equals

    • A. (2/3)(9 − x²)^(3/2) + C
    • B. (x/2)√(9 − x²) − (9/2) sin⁻¹(x/3) + C
    • C. (x/2)√(9 − x²) + (9/2) sin⁻¹(x/3) + C
    • D. (x/2)√(9 − x²) + (3/2) sin⁻¹(x/3) + C

26. Q26. ∫ dx/(sin x cos x) equals

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

27. Q27. ∫ x³/(x² + 1) dx equals

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

28. Q28. ∫ dx/(eˣ + 1) equals

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

29. Q29. ∫ cos 4x cos 2x dx equals

    • A. sin 6x/12 − sin 2x/4 + C
    • B. sin 6x/6 + sin 2x/2 + C
    • C. sin 6x/12 + sin 2x/4 + C
    • D. sin 8x/8 + C

30. Q30. ∫ √x ln x dx equals

    • A. (1/x^(1/2)) ln x + C
    • B. (2/3)x^(3/2) ln x − (4/9)x^(3/2) + C
    • C. (2/3)x^(3/2) ln x + (4/9)x^(3/2) + C
    • D. (2/3)x^(3/2) ln x − (2/3)x^(3/2) + C