CLASS 12 INTEGRATION DPP
INTEGRATION
✤ IMPORTANT QUESTION
1. Find ∫√tan x (1 + tan²x)dx
2. Find ∫3ax / (b² + c²x²) dx
3. Evaluate ∫e⁻¹+x⁻¹ / eˣ + x dx
4. Evaluate ∫sec⁴x tan x dx
5. Evaluate ∫(1-sin 2x) / (x + cos²x) dx
6. Evaluate ∫(x+1)(x+log x)² / x dx
7. Find ∫cot x . log(sin x) dx
8. Find ∫(1+tan x) / (1-tan x) dx
9. Find ∫(ex+3x)2 (ex +3) dx
✤ Long Answer I / Long Answer II
10. Find ∫ (sin x / (1-cos x)(2-cos x)) dx.
11. Evaluate ∫ (x / x4 - x2 + 1) dx.
12. Find ∫ (cos(x+a) / sin(x+b) dx.
13. Find ∫ (ex / √(5-4ex - e2x) dx.
14. Find ∫ (sin x + cos x) / 9+16 sin 2x dx.
15. Find ∫ tan³ x dx
16. Find ∫ (sin x) / (sin x + cos x) dx
17. Find ∫ (sin 2x) / (a cos² x + b sin² x) dx
18. Evaluate ∫ sin³ x dx
19. Evaluate ∫ 5(5x) . 5x . 5x dx
✤ PRACTICE QUESTION
❉ Very Short (Objective Type) / Short Answer Type
1. ∫sin⁶x/cos⁸x dx is equal to ______ .
2. ∫√x * e^(√x)/√x dx equals ______ .
3. The value of λ for which ∫(4x³+λ4x⁴)/(4x⁴+x⁴) dx = log|4x⁴ + x⁴| is
(a) 1 (b) logₑ4 (c) logₑ4 e (d) 4
Evaluate the following integrals in Exercises 4 to 10:
4. ∫cos√x/√x dx [NCERT; Foreign 2011]
5. ∫etan⁻¹x/(1+x²) dx [NCERT; AI 2011]
6. ∫1/((2-x)²+1) dx
7. ∫1/(x+x log x) dx [NCERT]
8. ∫(e²x-e⁻²x)/(e²x+e⁻²x) dx [NCERT; Foreign 2011]
9. ∫2x cos²x dx
10. ∫sin(tan⁻¹x)/(1+x²) dx [NCERT]
❆ Long Answer I / Long Answer II Type
Find the following integrals in Exercises 11 to 27:
11. ∫x^(1/2)/(x^(1/3)+x^(1/2)) dx [HOTS]
12. ∫cosec x * log (cosec x - cot x)dx
13. ∫f'(ax + b) [f(ax + b)]ⁿ dx
14. ∫dx/(1 + tan x) [NCERT]
15. ∫ (1 / √(sin³ x sin(x + α))) dx
16. ∫ ((x eˣ + 1)(eˣ + log x)²) / x dx
17. ∫ (1 + cot x) / (x + log(sin x)) dx
18. ∫ √eˣ - 1 dx
19. ∫ (sin x cos x) / (sin⁴ x + cos⁴ x) dx
20. ∫ dx / √(x + x)
21. ∫ tan⁴ x dx [NCERT]
22. ∫ (tan⁴ √x sec² √x) / √x dx [NCERT]
23. ∫ (x⁴ - x)¹/⁴ / x⁵ dx [NCERT]
24. ∫ (10x⁹ + 10ˣ logₑ 10) / (x¹⁰ + 10ˣ) dx [NCERT]
25. ∫ dx / (eˣ + e⁻ˣ) [NCERT]
26. ∫ √(1 + x²) / x⁴ dx [NCERT Exempler]
27. ∫ dx / (x³ (x⁵ + 1)³ᐟ⁵) [Guwahti 2015]
✤ INTEGRATION USING TRIGNOMETRIC IDENTITIES
1. Find ∫ 2 / (1 + cos 2x) dx
2. Find ∫ sec²x / cosec²x dx
3. Find ∫ (cosx - sinx)² dx
4. Evaluate ∫ √(1 + sin x/4) dx
✤ Long Answer I / Long Answer
5. Find ∫ tan⁻¹ √(1-sinx / 1+sinx) dx
6. Find ∫ sinx . sin2x . sin3x dx
7. Find ∫ cos 2x - cos 2α / cos x - cos α dx
8. Find ∫ sin(x - α) / sin(x + α) dx
9. Evaluate ∫ 1 - cos x / 1 + cos x dx
10. Evaluate ∫ sin⁴ 2x dx
11. Evaluate ∫ dx / 1-sinx
❉ Very Short (Objective Type) / Short Answer Type
1. If ∫ 1/√4-9x2 dx = 1/3 sin-1(ax) + C, then value of a is
2. Evaluate ∫ (1 / ((3+x)² + 1) ) dx
3. Evaluate ∫ (1 / √4 - x² ) dx
4. Evaluate ∫ (1 / 4 + 9x² ) dx
5. Find ∫ (dx / x² + 4x + 8)
6. Find ∫ (dx / 5 - 8x - x²)
7. Find ∫ (dx / √3 - 2x - x²)
6. Find ∫ (2x / (x²+1)(x²+2)²) dx
7. Find ∫ (2x / (x²+1)(x⁴+4) ) dx
8. Find ∫ (cosθ) / ((4 + sin²θ)(5 - 4cos²θ)) dθ
9. Find ∫ (sinθ) / ((4 + cos²θ)(2 - sin²θ)) dθ
10. Find ∫ ex / ((ex - 1)²(ex + 2)) dx
11. Find ∫ (ex / (2+ex)(4+ex)) dx
12. Find ∫ x / ((2+x2)(4+x2)) dx
13. Find ∫ (2 cos x / ((1-sin x)(1+sin2 x))) dx
14. Find ∫ (x2 / ((x2+1)(x2+4))) dx
✤ PRACTICE QUESTION
❉ Long Answer I / Long Answer II Type
Evaluate the following integrals in exercises 1 to 9 :
1. ∫ ((3x-1) / (x+2)2) dx [NCERT; Foreign 2012]
2. ∫ ((3x+5) / (x3-x2-x+1)) dx [NCERT; Delhi 2013 (C)]
3. ∫ (1 / (x3-1)) dx [HOTS]
4. ∫ (1 / (ex-1)) dx [NCERT]
5. ∫ cos x / (1 + sin x) (2 + sin x) dx [NCERT; Delhi 2019]
6. ∫ 1-x2 / x(1-2x) dx [NCERT; Foreign 2013]
7. ∫ x2-3x / (x-1)(x-2) dx
8. ∫ x2+1 / (x-1)2 (x+3) dx [AI 2012]
9. ∫ x2 / (x2+4)(x2+9) dx [DoE; Foreign 2015; Delhi 2013]
✤ Long Answer I / Long Answer II Type
Evaluate the following integrals in Exercises 1 to 10:
1. ∫ x sin⁻¹ x dx [Chennai 2015; HOTS]
2. ∫ tan⁻¹ (2x / (1-x²)) dx
3. ∫ x cot⁻¹ x dx
4. ∫ x²ⁿ⁻¹ cos xⁿ dx
5. ∫ x sin⁻¹ x² / √(1-x⁴) dx [HOTS]
6. ∫ (sin⁻¹√x - cos⁻¹√x) / (sin⁻¹√x + cos⁻¹√x) dx [NCERT; DoE]
7. ∫ (sin⁻¹x)² dx [NCERT]
8. ∫ sin⁻¹x / (1-x²)³/² dx [NCERT Exemplar]
9. ∫ √(1-√x) / (1+√x) dx [NCERT; DoE]
10. ∫ √(x²+1) [log(x²+1) - 2 log x] / x⁴ dx [NCERT; HOTS]
11. Find ∫ x³ / (x⁴ + x³) = 2 dx
12. Find∫ (x² + 1) ex / (x + 1)² dx
13. Find ∫ x√(x⁴ - 1) dx
14. Find 1 / a²sin²x + b²cos²x dx
15. Find ∫ √tan x dx
16. Find ∫ex (sin 4x-4)/(1-cos 4x) dx
17 . Find ∫e2x sin x dx
18. Find ∫e2x (1-sin 2x)/(1-cos 2x) dx
19. Find ∫ dx / (sin x + cos x)
20. Evaluate ∫ dx / (1 + 3 sin² x)
21. Find ∫ e²ˣ sin(3x + 1) dx
22. Evaluate ∫e³ˣ cos(3x - 5)dx
23. Evaluate ∫e⁵ˣ sin²x dx
24 ∫eˣ (1-x) / (1+x²)² dx is equal to
(a) eˣ / (1+x²) + C (b) -eˣ / (1+x²) + C
(c) eˣ / (1+x²)² + C (d) eˣ / (1-x²)² + C
25. ∫ (x-4)eˣ / (x-2)³ dx
26. ∫ 1 / [(x-1)√(x²+4)] dx
27. ∫ (2-x)eˣ / (1-x)² dx [HOTS]
28. ∫ dx / (cos²x + sin 2x) [HOTS]
29. ∫ (1 / (x-3)√x+1) dx
30. ∫ eax cos hx dx
31. ∫ ex (tan x - log cos x) dx
32. ∫ dx / 1-x^4
33. ∫ ex ((1-sin x) / (1-cos x)) dx
34. ∫ (x-3)√(x2+3x-18) dx
35. ∫ (tan x + log sec x) ex dx
36. ∫ x√1+x-x2 dx
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