Take the ECAT Mathematics — Integration (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…
The first 5 questions of this test, with the correct answer and explanation.
Q1. ∫ dx/(eˣ − e⁻ˣ), for x > 0, equals:
- A. ln(eˣ − e⁻ˣ) + C
- B. ½ ln[(eˣ − 1)/(eˣ + 1)] + C ✓
- C. ln[(eˣ − 1)/(eˣ + 1)] + C
- D. ½ ln[(eˣ + 1)/(eˣ − 1)] + C
Answer: B. ½ ln[(eˣ − 1)/(eˣ + 1)] + C
Explanation: Multiply by eˣ: ∫ eˣ/(e²ˣ − 1) dx = ∫ du/(u² − 1) with u = eˣ = ½ ln[(u − 1)/(u + 1)] + C.
Q2. ∫ dx/(x ln x ln(ln x)) equals:
- A. ln|ln x| + C
- B. 1/ln(ln x) + C
- C. ln|x ln x| + C
- D. ln|ln(ln x)| + C ✓
Answer: D. ln|ln(ln x)| + C
Explanation: Let u = ln(ln x), du = dx/(x ln x). The integral becomes ∫ du/u = ln|u| + C.
Q3. ∫ cos x/(sin x + cos x) dx equals:
- A. ½[x − ln|sin x + cos x|] + C
- B. ln|sin x + cos x| + C
- C. ½[x + ln|sin x + cos x|] + C ✓
- D. x + ln|sin x + cos x| + C
Answer: C. ½[x + ln|sin x + cos x|] + C
Explanation: Write cos x = ½[(cos x + sin x) + (cos x − sin x)]. The first part gives ½x and, since cos x − sin x is the derivative of sin x + cos x, the second gives ½ ln|sin x + cos x|.
Q4. ∫ √((1 − x)/(1 + x)) dx for −1 < x < 1 equals:
- A. sin⁻¹x + √(1 − x²) + C ✓
- B. ½ sin⁻¹x + √(1 − x²) + C
- C. cos⁻¹x + √(1 − x²) + C
- D. sin⁻¹x − √(1 − x²) + C
Answer: A. sin⁻¹x + √(1 − x²) + C
Explanation: Multiply top and bottom by √(1 − x): (1 − x)/√(1 − x²) = 1/√(1 − x²) − x/√(1 − x²). Integrating gives sin⁻¹x + √(1 − x²) + C.
Q5. ∫ dx/(x√(x² − 1)) for x > 1 equals:
- A. √(x² − 1) + C
- B. cos⁻¹x + C
- C. ln(x + √(x² − 1)) + C
- D. sec⁻¹x + C ✓
Answer: D. sec⁻¹x + C
Explanation: With x = sec θ the integral becomes ∫ dθ = θ = sec⁻¹x + C.
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Q6. ∫ e²ˣ sin 3x dx equals:
- A. e²ˣ(3 sin 3x − 2 cos 3x)/13 + C
- B. e²ˣ(2 sin 3x − 3 cos 3x)/5 + C
- C. e²ˣ(2 sin 3x − 3 cos 3x)/13 + C
- D. e²ˣ(2 sin 3x + 3 cos 3x)/13 + C
Q7. ∫ x³ e^(x²) dx equals:
- A. ½ e^(x²)(x² − 1) + C
- B. ½ e^(x²)(x² + 1) + C
- C. e^(x²)(x² − 1) + C
- D. ¼ x⁴ e^(x²) + C
Q8. ∫ e^(√x) dx for x > 0 equals:
- A. 2(√x − 1)e^(√x) + C
- B. (√x − 1)e^(√x) + C
- C. 2e^(√x) + C
- D. 2(√x + 1)e^(√x) + C
Q9. ∫ x/(1 + cos x) dx on (−π, π) equals:
- A. x tan(x/2) + ln cos(x/2) + C
- B. x cot(x/2) + 2 ln sin(x/2) + C
- C. x tan(x/2) − 2 ln cos(x/2) + C
- D. x tan(x/2) + 2 ln cos(x/2) + C
Q10. ∫ dx/(x(x⁵ + 1)) equals:
- A. (1/5) ln|x⁵ + 1| − ln|x| + C
- B. ln|x⁵ + 1| + C
- C. ln|x| − (1/5) ln|x⁵ + 1| + C
- D. ln|x| − ln|x⁵ + 1| + C
Q11. ∫ (3x + 1)/√(x² + 2x + 5) dx equals:
- A. 3√(x² + 2x + 5) − 2 ln|x + 1 + √(x² + 2x + 5)| + C
- B. 3√(x² + 2x + 5) + 2 ln|x + 1 + √(x² + 2x + 5)| + C
- C. 3√(x² + 2x + 5) − 2 sin⁻¹((x + 1)/2) + C
- D. (3/2)√(x² + 2x + 5) − 2 ln|x + 1 + √(x² + 2x + 5)| + C
Q12. ∫ dx/√(3 − 2x − x²) equals:
- A. sin⁻¹((x − 1)/2) + C
- B. sin⁻¹((x + 1)/2) + C
- C. sin⁻¹(x + 1) + C
- D. cos⁻¹((x + 1)/2) + C
Q13. ∫ dx/(x³ + x) equals:
- A. ln|x| − ½ ln(x² + 1) + C
- B. ln|x| + ½ ln(x² + 1) + C
- C. ½ ln|x| − ln(x² + 1) + C
- D. ln|x| − tan⁻¹x + C
Q14. ∫ dx/((x + 1)(x² + 1)) equals:
- A. ln|x + 1| − ½ ln(x² + 1) + tan⁻¹x + C
- B. ½ ln|x + 1| − ½ ln(x² + 1) + ½ tan⁻¹x + C
- C. ½ ln|x + 1| − ¼ ln(x² + 1) + ½ tan⁻¹x + C
- D. ½ ln|x + 1| − ¼ ln(x² + 1) − ½ tan⁻¹x + C
Q15. ∫ sin³x cos²x dx equals:
- A. sin⁴x cos³x/12 + C
- B. −cos³x/3 + cos⁵x/5 + C
- C. sin³x/3 − sin⁵x/5 + C
- D. cos³x/3 − cos⁵x/5 + C
Q16. ∫ sin 2x/(3 + cos²x) dx equals:
- A. −ln(3 + sin²x) + C
- B. −2 ln(3 + cos²x) + C
- C. −ln(3 + cos²x) + C
- D. ln(3 + cos²x) + C
Q17. ∫ eˣ/(e²ˣ − 3eˣ + 2) dx equals:
- A. ln|(eˣ − 1)/(eˣ − 2)| + C
- B. ln|(eˣ − 2)/(eˣ − 1)| + C
- C. ln|(eˣ − 2)(eˣ − 1)| + C
- D. ln|eˣ − 2| + ln|eˣ − 1| + C
Q18. ∫ x³/√(1 − x⁸) dx equals:
- A. sin⁻¹(x⁴) + C
- B. −½√(1 − x⁸) + C
- C. ¼ sin⁻¹(x²) + C
- D. ¼ sin⁻¹(x⁴) + C
Q19. If f′(x) = x/√(x² + 9) and f(4) = 7, then f(0) equals:
Q20. A particle has acceleration a = 6t (m/s²), initial velocity 2 m/s and initial position 1 m. Its position at t = 2 s is:
- A. 17 m
- B. 13 m
- C. 11 m
- D. 9 m
Q21. ∫ (√x + 1/√x)² dx for x > 0 equals:
- A. x²/2 + 2x + ln x + C
- B. x²/2 + x + ln x + C
- C. x²/2 + 2 ln x + C
- D. x²/2 + 2x − 1/x + C
Q22. ∫ sin(ln x) dx equals:
- A. −x cos(ln x) + C
- B. (x/2)[sin(ln x) − cos(ln x)] + C
- C. (x/2)[sin(ln x) + cos(ln x)] + C
- D. (x/2)[cos(ln x) − sin(ln x)] + C
Q23. ∫ (ln x)/x² dx for x > 0 equals:
- A. −(ln x + 1)/x + C
- B. −ln x/x + C
- C. (ln x)²/(2x²) + C
- D. −(ln x − 1)/x + C
Q24. ∫ dx/(1 − sin x) equals:
- A. cot x − cosec x + C
- B. ln|1 − sin x| + C
- C. tan x + sec x + C
- D. tan x − sec x + C
Q25. ∫ x²/(1 + x²)² dx equals:
- A. ½[tan⁻¹x + x/(1 + x²)] + C
- B. −1/(1 + x²) + C
- C. tan⁻¹x − x/(1 + x²) + C
- D. ½[tan⁻¹x − x/(1 + x²)] + C
Q26. ∫ 2ˣ eˣ dx equals:
- A. 2ˣ eˣ/(ln 2) + C
- B. (2e)ˣ/(1 + ln 2) + C
- C. (2e)ˣ + C
- D. 2ˣ eˣ/e + C
Q27. ∫ eˣ(1 + sin x)/(1 + cos x) dx equals:
- A. eˣ tan x + C
- B. eˣ cot(x/2) + C
- C. eˣ sec²(x/2) + C
- D. eˣ tan(x/2) + C
Q28. ∫ x cosh x dx equals:
- A. x sinh x − cosh x + C
- B. x cosh x − sinh x + C
- C. ½x² sinh x + C
- D. x sinh x + cosh x + C
Q29. ∫ tan³x dx equals:
- A. ½tan²x + ln|cos x| + C
- B. ½tan²x − ln|cos x| + C
- C. tan⁴x/4 + C
- D. ½tan²x + ln|sin x| + C
Q30. ∫ dx/(x√(1 + ln x)) equals:
- A. √(1 + ln x) + C
- B. ln|1 + ln x| + C
- C. 2√(1 + ln x) + C
- D. (2/3)(1 + ln x)^(3/2) + C