Indefinite Integrals Class 12: Formulas, Rules & Solved Examples

Indefinite Integrals Class 12: Formulas, Rules & Solved Examples

If differentiation asks "what is the rate of change of this function?", integration asks the opposite: "given the rate of change, what was the original function?" This reverse process is called anti-differentiation, and it forms the foundation of Chapter 7 (Integrals) in CBSE Class 12 Maths. Whether you're solving board problems or tackling quant sections in CUET, IPMAT, or NPAT, a solid grip on indefinite integrals is non-negotiable. This guide breaks down the formulas, properties, and techniques you need — with worked examples.

What Is an Indefinite Integral?

An indefinite integral represents the entire family of anti-derivatives of a function. If F(x) is a function such that F′(x) = f(x), then F(x) is called an anti-derivative (or integral) of f(x). Since the derivative of any constant is zero, F(x) + C is also a valid anti-derivative for any real number C.

We write this as:

∫f(x) dx = F(x) + C

Here, f(x) is the integrand, x is the variable of integration, and C is the constant of integration. The ∫ symbol connects indefinite integrals to definite integrals and area calculations later in the chapter.

Why Do We Always Add "+ C"?

This is one of the most commonly misunderstood ideas in the chapter. Since differentiating a constant always gives zero, infinitely many functions share the same derivative — they just differ by a constant. For example, sin x + 1, sin x + 5, and sin x − 3 all have the same derivative, cos x. So when we integrate cos x, we can't pin down one specific function; instead, we represent the entire family using sin x + C.

This is more than a technicality — examiners routinely deduct marks for a missing "+ C" in board exams, so make it a habit from day one.

Standard Integration Formulas You Must Memorize

These formulas are derived directly from standard derivative results, and they form the backbone of every integration problem you'll face:

Notice the pattern: each formula is simply the reverse of a familiar differentiation rule. Instead of memorizing them as isolated facts, try deriving each one from its corresponding derivative — it sticks better and helps during revision.

Key Properties of Indefinite Integrals

Beyond the standard formulas, five core properties let you break complicated integrals into manageable pieces:

1. Differentiation and integration are inverse processes: d/dx[∫f(x)dx] = f(x), and ∫f′(x)dx = f(x) + C.
2. Two integrals with the same derivative are equivalent — they belong to the same family of curves, differing only by a constant.
3. Sum rule: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
4. Constant multiple rule: ∫k·f(x)dx = k∫f(x)dx, for any real number k
5. These generalize to any finite number of functions and constants.

These let you split a messy expression into simpler standard integrals.

Solved Examples

Example 1: Find ∫(x³ − 1)/x² dx

Split the fraction first: ∫(x − x⁻²) dx = ∫x dx − ∫x⁻² dx = x²/2 − (x⁻¹/−1) + C = x²/2 + 1/x + C

Example 2: Find ∫(sin x + cos x) dx

Using the sum rule directly: ∫sin x dx + ∫cos x dx = −cos x + sin x + C

Example 3: Find the anti-derivative F of f(x) = 4x³ − 6, given F(0) = 3.

First find the general anti-derivative: ∫(4x³ − 6)dx = x⁴ − 6x + C. Substituting F(0) = 3: 0 − 0 + C = 3, so C = 3. Therefore, F(x) = x⁴ − 6x + 3.

This shows a favorite board-exam question type — finding a specific anti-derivative using an initial condition, rather than the general family.

Common Mistakes Students Make

• Dropping the "+ C": examiners deduct marks even if the rest of your working is correct.
• Misapplying the power rule at n = −1: ∫xⁿdx = xⁿ⁺¹/(n+1)+C fails here; use ∫(1/x)dx = log|x|+C instead.
• Forgetting the modulus in log|x|: log is undefined for negative numbers.
• Not simplifying first: many problems (like Example 1) become trivial once split or simplified.

Quick Tips for CBSE Boards & CUET Preparation

CBSE board papers often test formula recall combined with algebraic manipulation — expanding brackets, splitting fractions, or using basic identities before applying a standard formula. For CUET and other entrance exams, speed matters as much as accuracy, so drill formula recall until it's automatic.

Suggested Internal Links

• Link to: Integration by Substitution Method – Class 12 Maths (next logical step after mastering standard formulas)
• Link to: Definite Integrals & Fundamental Theorem of Calculus (for students ready to apply indefinite integrals to area problems)
• Link to: CUET Maths Preparation Guide (for entrance-exam-focused readers)
• Link to: CBSE Class 12 Maths Chapter-wise Question Bank (practice resource)

Ready to Master Integration?

Formulas are only half the battle — real exam confidence comes from solving enough varied problems until the patterns become second nature. On ChampionsPrep, you get AI-powered doubt resolution and topic-wise practice questions for just ₹10 per use, so you can clear every integration doubt the moment it comes up, without waiting for the next tuition class. Start practicing Indefinite Integrals on ChampionsPrep →

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