ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Integration (Set 5)

Take the ECAT Mathematics — Integration (Set 5): 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. ∫ x/√(x + 1) dx equals:

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

   Answer: A. 2(x + 1)^(3/2)/3 − 2√(x + 1) + C

   Explanation: Put u = x + 1: ∫ (u − 1)u^(−1/2) du = (2/3)u^(3/2) − 2u^(1/2).

2. Q2. ∫ dx/[x(x + 1)²] (x > 0) equals:

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

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

   Explanation: 1/[x(x + 1)²] = 1/x − 1/(x + 1) − 1/(x + 1)², so the integral is ln x − ln(x + 1) + 1/(x + 1).

3. Q3. ∫ sin x cos x/(1 + sin²x) dx equals:

   • A. ln(1 + sin²x) + C
   • B. tan⁻¹(sin x) + C
   • C. ln(1 + sin²x)/2 + C ✓
   • D. −ln(1 + sin²x)/2 + C

   Answer: C. ln(1 + sin²x)/2 + C

   Explanation: Put u = 1 + sin²x, du = 2 sin x cos x dx: ½ ln u.

4. Q4. ∫ eˣ/√(1 − e²ˣ) dx equals:

   • A. sin⁻¹ eˣ + C ✓
   • B. √(1 − e^(2x)) + C
   • C. cos⁻¹ eˣ + C
   • D. tan⁻¹ eˣ + C

   Answer: A. sin⁻¹ eˣ + C

   Explanation: Put u = eˣ: ∫ du/√(1 − u²) = sin⁻¹u.

5. Q5. ∫ x²/(1 + x⁶) dx equals:

   • A. tan⁻¹(x³) + C
   • B. tan⁻¹(x³)/9 + C
   • C. ln(1 + x⁶)/6 + C
   • D. tan⁻¹(x³)/3 + C ✓

   Answer: D. tan⁻¹(x³)/3 + C

   Explanation: Put u = x³, du = 3x² dx: (1/3)∫ du/(1 + u²) = (1/3)tan⁻¹(x³).

The other 25 questions

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6. Q6. ∫ x sec x tan x dx on (0, π/2) equals:

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

7. Q7. ∫ ln(x² + 1) dx equals:

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

8. Q8. ∫ sec⁴x dx equals:

   • A. tan x + tan³x + C
   • B. tan³x/3 + C
   • C. tan x + tan³x/3 + C
   • D. sec⁵x/5 + C

9. Q9. ∫ dx/(1 − cos x) equals:

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

10. Q10. ∫ x e^(2x) dx equals:

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

11. Q11. ∫ x³e^(x²) dx equals:

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

12. Q12. ∫ dx/[x√(x² − 1)] for x > 1 equals:

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

13. Q13. ∫ (x + 2)/(x² + 4x + 7) dx equals:

    • A. ln(x² + 4x + 7) + C
    • B. ln(x² + 4x + 7)/4 + C
    • C. tan⁻¹((x + 2)/√3)/√3 + C
    • D. ln(x² + 4x + 7)/2 + C

14. Q14. ∫ dx/(x² + x + 1) equals:

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

15. Q15. ∫ 2ˣ·3ˣ dx equals:

    • A. 6ˣ/ln 6 + C
    • B. 6ˣ + C
    • C. x 6^(x − 1) + C
    • D. 6ˣ ln 6 + C

16. Q16. ∫ dx/[√x(1 + √x)] equals:

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

17. Q17. ∫ sin⁻¹x/√(1 − x²) dx equals:

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

18. Q18. ∫ e⁻ˣ cos x dx equals:

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

19. Q19. ∫ x/(1 + x⁴) dx equals:

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

20. Q20. ∫ sinh²x dx equals:

    • A. sinh 2x/2 − x/2 + C
    • B. sinh 2x/4 + x/2 + C
    • C. sinh³x/3 + C
    • D. sinh 2x/4 − x/2 + C

21. Q21. ∫ cos x/√(1 + sin x) dx equals:

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

22. Q22. If ∫ f(x) dx = g(x) + C, then ∫ x f(x²) dx equals:

    • A. ½ g(x²) + C
    • B. g(x²) + C
    • C. ½ x g(x²) + C
    • D. 2x g(x²) + C

23. Q23. ∫ x³/(x + 1) dx equals:

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

24. Q24. ∫ sec x (sec x + tan x) dx equals:

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

25. Q25. ∫ cos 2x/(sin²x cos²x) dx equals:

    • A. −cot x + tan x + C
    • B. −cot x − tan x + C
    • C. cot x + tan x + C
    • D. cot x − tan x + C

26. Q26. ∫ dx/√(2x − x²) equals:

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

27. Q27. ∫ x√(1 − x) dx equals:

    • A. x² √(1 − x)/2 + C
    • B. −2(1 − x)^(3/2)/3 − 2(1 − x)^(5/2)/5 + C
    • C. 2(1 − x)^(3/2)/3 − 2(1 − x)^(5/2)/5 + C
    • D. −2(1 − x)^(3/2)/3 + 2(1 − x)^(5/2)/5 + C

28. Q28. ∫ ln(e^(x²)) dx equals:

    • A. 2x³/3 + C
    • B. x³/3 + C
    • C. x²/2 + C
    • D. x³ + C

29. Q29. ∫ sin(ln x)/x dx (x > 0) equals:

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

30. Q30. The function F with F′(x) = x eˣ and F(0) = 0 is:

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