Currently Empty: $0.00
5.Indefinite Integrals
Class 12 • Mathematics • NCERT
Indefinite Integrals
[ Embed Your Video Lecture Here ]
This lesson introduces the concept of indefinite integrals as the inverse process of differentiation, following the NCERT Class 12 syllabus.
Lesson Objectives
- Understand integration as inverse of differentiation.
- Learn standard integrals.
- Apply basic integration formulas.
- Solve NCERT exam-based problems.
1. Meaning of Indefinite Integral
An indefinite integral represents a family of functions whose derivative is the given function.
∫ f(x) dx = F(x) + C
2. Constant of Integration
The constant C appears because the derivative of a constant is zero.
d/dx (F(x) + C) = f(x)
3. Standard Integrals (NCERT)
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C , n ≠ −1
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
4. Properties of Indefinite Integrals
Indefinite integrals satisfy linearity properties.
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫ kf(x) dx = k ∫ f(x) dx
∫ kf(x) dx = k ∫ f(x) dx
5. NCERT Solved Example
Evaluate: ∫ (3x² + 4x) dx
= 3 ∫ x² dx + 4 ∫ x dx
= 3 (x³/3) + 4 (x²/2) + C
= x³ + 2x² + C
6. Important NCERT Notes
• Indefinite integral represents a family of curves
• Constant of integration is compulsory
• Integration is inverse of differentiation
• Used in definite integrals and differential equations
• Constant of integration is compulsory
• Integration is inverse of differentiation
• Used in definite integrals and differential equations
Practice Questions (NCERT)
- Define indefinite integral.
- What is constant of integration?
- Find ∫ x² dx.
- Find ∫ cos x dx.
- Evaluate ∫ (2x + 3) dx.
- Write ∫ 1/x dx.
- Is constant of integration necessary?
- Find ∫ eˣ dx.
- State linearity property of integrals.
- What is ∫ 0 dx?
✅ Show Answer Key
- Family of functions whose derivative is given function
- Arbitrary constant C
- x³/3 + C
- sin x + C
- x² + 3x + C
- ln|x| + C
- Yes
- eˣ + C
- ∫(f+g)=∫f+∫g
- C
© Aviate Learning – Indefinite Integrals (NCERT Class 12)