📐 Integration — Practice Worksheet

Q1. ∫ x³ dx equals:

1. x⁴ + C
2. x⁴/4 + C
3. 3x² + C
4. 4x⁴ + C

Q2. ∫ (1/x) dx equals:

1. x⁻² + C
2. ln|x| + C
3. −1/x² + C
4. eˣ + C

Q3. ∫₀^π sin x dx equals:

1. 0
2. 1
3. 2
4. −1

Q4. ∫ eˣ(1 + eˣ)⁻¹ dx equals:

1. ln(1 + eˣ) + C
2. eˣ ln(1 + eˣ) + C
3. (1 + eˣ)⁻² + C
4. eˣ/(1 + eˣ) + C

Q5. ∫ sec²x dx equals:

1. sec x tan x + C
2. tan x + C
3. −cot x + C
4. sin x / cos²x + C

Q6. ∫₀¹ x eˣ dx equals:

1. 1
2. e − 1
3. e
4. e + 1

Q7. ∫ sin²x dx equals:

1. (x − sin 2x / 2)/2 + C
2. (x/2) − (sin 2x)/4 + C
3. −cos²x + C
4. sin x cos x + C

Q8. ∫ dx / (x² + a²) equals:

1. (1/a) tan⁻¹(x/a) + C
2. ln(x² + a²) + C
3. sin⁻¹(x/a) + C
4. x/(x² + a²) + C

Q9. If ∫₀^a f(x) dx = ∫₀^a f(a − x) dx, this property is known as:

1. King's rule
2. Leibniz rule
3. Fundamental theorem
4. Mean value theorem

Q10. ∫ dx / √(1 − x²) equals:

1. tan⁻¹x + C
2. sin⁻¹x + C
3. cos⁻¹x + C
4. sec⁻¹x + C

Q11. ∫₋ₐ^a f(x) dx = 0 when f(x) is:

1. An even function
2. An odd function
3. A periodic function
4. A constant function

Q12. ∫ x sin x dx equals:

1. −x cos x + sin x + C
2. x cos x − sin x + C
3. −x cos x − sin x + C
4. x sin x + cos x + C

Q13. The value of ∫₀^(π/2) log(tan x) dx is:

1. 0
2. π/4
3. 1
4. π/2

Q14. ∫ eˣ(sin x + cos x) dx equals:

1. eˣ sin x + C
2. eˣ cos x + C
3. eˣ(sin x − cos x) + C
4. eˣ(sin x + cos x) + C

Q15. ∫ dx / (x² − a²) equals:

1. (1/2a) ln|(x − a)/(x + a)| + C
2. (1/a) tan⁻¹(x/a) + C
3. ln|x² − a²| + C
4. (1/2a) ln|(x + a)/(x − a)| + C

Q16. ∫₀^1 dx / (1 + x²) equals:

1. π/2
2. π/4
3. π/3
4. 1

Q17. ∫ (2x + 3) / (x² + 3x + 5) dx equals:

1. ln|x² + 3x + 5| + C
2. (x² + 3x + 5)⁻¹ + C
3. 2 ln|x² + 3x + 5| + C
4. tan⁻¹(x² + 3x + 5) + C

Q18. The area bounded by y = x², the x-axis, and the lines x = 0 and x = 2 is:

1. 4/3
2. 8/3
3. 4
4. 2

Q19. Evaluate: ∫ (3x² + 2x − 1) / (x³ + x² − x) dx using partial fractions.

Q20. Evaluate: ∫₀^(π/2) sin⁴x cos²x dx using reduction formulas or Wallis' formula.

Q21. Find the area enclosed between the curves y = x² and y = 2x − x².

Q22. Evaluate: ∫ x² eˣ dx using integration by parts.

Q23. Evaluate: ∫ dx / (sin x + cos x). (Hint: use the substitution t = tan(x/2))

Q24. Evaluate: ∫₀^π x sin x / (1 + cos²x) dx using the property ∫₀^a f(x) dx = ∫₀^a f(a−x) dx.

Q25. Find the area bounded by the ellipse x²/a² + y²/b² = 1 using integration.

Q26. Evaluate: ∫ √(x² + 4x + 8) dx by completing the square and using appropriate substitution.

Q27. Evaluate: ∫₀^∞ e⁻ˣ² dx (state the result and explain its significance — Gaussian integral).

Q28. Evaluate: ∫ (x + 1) / (x² + 1)² dx using the substitution x = tan θ.

Q29. Find the volume of the solid generated by revolving y = √x from x = 0 to x = 4 about the x-axis.

Q30. Evaluate: ∫₀^1 ln(1 + x) / (1 + x²) dx. (Hint: consider substitution x = tan θ)