Download as PDF, PPTX
instegration basic notes of class 12th h
- 1.
- 2. INTEGRATION JEE MAINS &ADVANCED COURSE ➔ Foundation Sessions for starters ➔ Complete PYQ’s (2015-2023) ➔ NTA + Cengage CHAPTER WISE Questions ➔ My HANDWRITTEN Notes tinyurl.com/jeewithnehamam WE DO NOT SELL ANY COURSES For FREE & Focused JEE MATERIAL, CLICK to Join TELEGRAM : t.me/mathematicallyinclined
- 3. CLICK HERE FORTHE FULL PLAYLIST S. No. TOPICS 1. Logarithms 2. Inequalities - Wavy Curve 3. Trigonometry 4. Functions: Basics, Domain & Range 5. Modulus & Greatest Integer Functions 6. Complex Numbers 7. Differentiation 8. Limits 9. Quadratic Equations 10. Permutations & Combinations S. No. TOPICS 11. Probability 12. Binomial Theorem 13. Integration 14. Sequence & Series 15. Vectors 16. 3D Geometry 17. Graphs & Transformation 18. Straight Lines & Circles 19. Sets & Relations 20. Conic Sections 21. Matrices and Determinants
- 4. INTEGRATION ● What isIntegration ● Integration Notation ● Types of Integrations ● Constant of Integration ● Basic Integration Formulae ● Properties of Indefinite Integration ● Integration by Substitution ● Integration by Parts ● Some Standard Integration ● Definite Integration ● Geometrical Meaning of Integration INTEGRATION BASIC
- 5. INTEGRATION JEE(MAIN) PYQs UsingINTEGRATION
- 6.
- 7. INTEGRATION Indefinite integration isthe reverse process of differentiation Hence it is also called as antiderivative What is Integration
- 8. INTEGRATION Integration Notation Integration isdenoted by an integral sign ∫. y = ∫f(x) dx+ = F(x) + c Constant of Integration Integrand Variable of integration F ’(x) also = f(x) (First derivative)
- 9. INTEGRATION Types of Integrations DefiniteIntegration Indefinite Integration
- 10. INTEGRATION Indefinite Integration asThe Reverse Process of Differentiation Derivative Integral Sin (x) + C Cos (x) d dx {F(x)} = f(x) then, ∫ f(x) dx = F(x) + c If
- 11. INTEGRATION e.g. Similarly, d dx (x2) = 2xso ∫ 2x dx = x2 + c d dx (sin x) = cos x so ∫ cos x dx = sin x + c Why c? Indefinite Integration as The Reverse Process of Differentiation
- 12. INTEGRATION d dx (x2) = 2x d dx (x2+ 1) = 2x d dx (x2 + 2) = 2x . . . . ∫ 2x dx = x2 + c c = 0 c = 1 c = 2 . . where, c = constant of integration Thus, the general value of ∫ f(x) dx = F(x) + c d dx (x2 + c) = 2x Constant of Integration
- 13. INTEGRATION Constant of Integration:Graphical Approach
- 14. INTEGRATION Q. What isAntiderivative of x5 ? A B C D x6+c 5x4+c
- 15.
- 16. INTEGRATION A B C D x6+c 5x4+c Q. What isAntiderivative of x5 ?
- 17. INTEGRATION Q. What is? A B C D x4+sin(x)+c x4-sin(x)+c 12x2+sin(x)+c 12x2- sin(x)+c
- 18.
- 19. INTEGRATION Q. What is? A B C D x4+sin(x)+c x4-sin(x)+c 12x2+sin(x)+c 12x2- sin(x)+c
- 20. INTEGRATION = x +c dx ∫xn dx = xn+1 n + 1 + c (n ≠ –1) d dx (x) = 1. 1 d dx xn+1 n + 1 = 2. xn ∫ 3. = x 1 x dx 1 =log x + c d dx (log x ) ∫ Basic Integration Formulae
- 21. INTEGRATION d dx ex = ex ex dx= ex + c 4. ∫ d dx ax log a = ax ∫ax dx = ax log a + c 5. Basic Integration Formulae
- 22. INTEGRATION d dx (–cos x) =sinx ∫sin x dx = – cos x + c d dx (sin x) = cos x ∫ dx cos x = sin x + c d dx (tan x) ∫ dx sec2 x = tan x + c = sec2 x 7. 8. 6. Basic Integration Formulae
- 23. INTEGRATION d dx (– cot x)= cosec2 x ∫cosec2 x dx = – cot x + c 9. d dx sec x = sec x tan x ∫sec x tan x dx = sec x + c 10. d dx (– cosec x) = cosec x cot x ∫ cosec x cot x dx =– cosec x + c 11. Basic Integration Formulae
- 24. INTEGRATION Basic Integration Formulae:Summary 6. 7. 8. 9. 10. 11.
- 25. INTEGRATION Properties of IndefiniteIntegration ∫ k f(x) dx = k ∫ f(x) dx where k is a constant 2. 1. ∫ (f1(x) + f2(x)) dx = ∫ f1(x) dx + ∫ f2(x) dx 3.
- 26.
- 27.
- 28.
- 29.
- 30.
- 31.
- 32.
- 33.
- 34. INTEGRATION Q. Write thevalue of
- 35.
- 36.
- 37.
- 38. INTEGRATION Q. i) ∫tan2x dx ii) ∫ cot2x dx
- 39. INTEGRATION Solution: We know (1+ tan2 x = sec2 x) tan2x = sec2 x –1 ∴ ∫(sec2 x– 1)dx ∴I = tan x – x + c = ∫ tan2 x dx i) ∫ cot2 x dx ii) We know (1 + cot2 x = cosec2 x) cot2x = cosec2 x –1 ∴ ∫(cosec2 x – 1)dx ∴I = – cot x – x + c =
- 40. INTEGRATION Q. ∫ cos2x 4– 5 sin x dx
- 41. INTEGRATION Solution: ∫ cos2x 4 –5 sin x dx ∫ 4 cos2x – 5 sin x cos2x dx = ∫(4 sec2 x – 5 sec x tan x) dx = We know ∫ sec2x dx = tan x ∫ & sec x tan x dx = sec x = 4 tan x – 5 sec x + c
- 42. INTEGRATION tan x cotx + C tan x - cot 2x + C tan x + cot x + C tan x - cot x + C Q. A B D C A B D C D
- 43.
- 44. INTEGRATION tan x cotx + C tan x - cot 2x + C tan x + cot x + C tan x - cot x + C Q. A B D C A B D C D
- 45. INTEGRATION Q. ∫x3 –x2 + x – 1 x – 1 dx
- 46. INTEGRATION ∫ (x –1) (x2 + 1) (x – 1) dx = = 3 x3 + x + c ∫(x2 + 1) dx = Solution:
- 47.
- 48.
- 49.
- 50. INTEGRATION Steps for Integrationby Substitution 1. Choose an appropriate function to substitute whose derivative will replace the other terms of the Integral. 2. Determine the value of dx. 3. Substitute the integral 4. Integrate the resulting function. 5. Return to the initial variable.
- 51. INTEGRATION I = ×g′ (x) dx f (g (x) ) ∫ Put g (x) = t g′ (x) dx = dt Differentiate w.r.t. x I = f (t) dt ∫ ∴
- 52. INTEGRATION Q. Integrate: ∫exsin(ex)dx
- 53.
- 54. INTEGRATION Q. Integrate: ∫2xsin(x2 + 1)dx
- 55. INTEGRATION We know thatthe derivative of (x2 + 1) = 2x. Hence, let’s substitute (x2 + 1) = t, so that 2x.dx = dt. Therefore,
- 56.
- 57.
- 58. INTEGRATION Q. Integrate thefunction :
- 59.
- 60.
- 61.
- 62. INTEGRATION Instead of solvingentire sum, just remember the shortcuts ∫ f n (x) f′(x) dx 1) Special Cases of Substitution
- 63. INTEGRATION ∫ tn dt I= = 1 n + 1 f n+1 (x) + c Put f (x) t = ∫f n (x) f′(x) dx 1) f′(x)dx dt = ∫ f n (x) f′(x) dx = 1 n + 1 f n+ 1 (x) + c Special Cases of Substitution Shortcut
- 64.
- 65.
- 66. INTEGRATION f ′(x) f (x) ∫dx 2) Special Cases of Substitution
- 67. INTEGRATION Put f (x)t = f ′(x) dx dt = 1 t ∫ dt I = ln t + c = ln f (x) + c = ∫ f ′(x) f(x) ln dx = f (x) + c f ′(x) f (x) ∫ dx 2) Special Cases of Substitution Shortcut
- 68. INTEGRATION Q. Integrate thefunction :
- 69.
- 70. INTEGRATION ∫ f ′(x) f(x) dx 3) √ Special Casesof Substitution
- 71. INTEGRATION = 2√t +2 = 2 f(x) + c √ Put f (x) t = f ′(x) dx dt = 1 t ∫ dt I = √ ∫ f ′(x) f(x) dx 3) √ Special Cases of Substitution ∫ f ′(x) f(x) dx √ Shortcut
- 72.
- 73.
- 74. INTEGRATION d dx (sin–1 x) = 1 1– x2 √ ∫ 1 1 – x2 √ dx = sin–1 x + c d dx (tan–1 x) = 1 1 + x2 ∫ 1 1 + x2 dx = tan–1 x + c Some Other Integrals
- 75.
- 76.
- 77.
- 78.
- 79.
- 80.
- 81.
- 82. INTEGRATION I L A T E INVERSE LOGARITHMIC ALGEBRAIC TRIGONOMETRIC EXPONENTIAL ⮚ To solveIntegration by Parts we make use of ILATE rule ∫f(x) g(x) = – f(x) ∫g(x)dx d dx f(x) ×∫g(x)dx dx ∫
- 83.
- 84.
- 85.
- 86. INTEGRATION = ln x f(x) g (x) = 1 ∫ln x × 1 dx = x ln x – x + c = ln (x) x – ∫ x 1 × x dx = x ln (x) – ∫ 1 dx Solution:
- 87.
- 88. INTEGRATION f(x) = x2 g(x)= cos2x I sin 2x = x2 2 –∫2x × sin 2x 2 dx = x2 sin 2x 2 – x (–cos 2x ) 2 –∫(–cos 2x ) 2 dx sin 2x sin 2x = x2 2 – –x cos 2x 2 + 4 = x2 sin 2x 2 + x cos 2x 2 – sin 2x 4 + c Solution:
- 89. INTEGRATION Some standard integrals dx x2+ a2 ∫ 1 a tan–1 x a + c = dx a2 – x2 ∫ 1 2a ln a + x a – x + c = dx x2 – a2 ∫ = 1 2a ln + c x - a x + a • • • dx a2 – x2 ∫ dx x2 – a2 ∫ = _
- 90. INTEGRATION = ln +c x + • ∫ dx x2 + a2 √ • ∫ dx a2 – x2 √ sin–1 x a + c = x2 + a2 √ = ln + c x + • ∫ dx x2 - a2 √ x2 - a2 √ Some standard integrals
- 91.
- 92.
- 93.
- 94.
- 95.
- 96. INTEGRATION Geometrical Meaning ofIntegration Definite Integrations Indefinite Integrations Unbounded Area Bounded Area
- 97.
- 98.
- 99.
- 100.
- 101.
- 102. INTEGRATION JEE MAINS &ADVANCED COURSE ➔ Foundation Sessions for starters ➔ Complete PYQ’s (2015-2023) ➔ NTA + Cengage CHAPTER WISE Questions ➔ My HANDWRITTEN Notes tinyurl.com/jeewithnehamam WE DO NOT SELL ANY COURSES For FREE & Focused JEE MATERIAL, CLICK to Join TELEGRAM : t.me/mathematicallyinclined