NCERT Class 12 Integrals: Formulas and Examples

Ever looked at a derivative and wondered, “How do I get the original function back?” That’s exactly what integrals help you do. While differentiation breaks a function into smaller parts, integration rebuilds it step by step. In Class 12, integrals might feel confusing at first—but once you understand the basics, they become much easier and even interesting. From finding areas under curves to solving real-life physics problems, integrals are widely used. In this guide, you’ll learn integrals in a simple, structured way—with formulas, clear explanations, and plenty of practice questions.

What are Integrals? 

Integrals are the inverse (opposite) of differentiation. This means if differentiation helps us find the rate at which something is changing, integration helps us go back and find the original function.

In mathematical terms:

If the derivative of a function is , then the integral of gives us back .

For example:
If

Then reversing it:

Why do we add “+ C”?

Whenever we differentiate a constant, the result is zero. So, if we only know the derivative, we cannot tell what constant was originally present. That’s why we always add + C, called the constant of integration.

Mini Examples:

Real-Life Understanding:

Think of differentiation like breaking something into smaller parts, and integration like rebuilding it completely.

Another way to understand integrals is through area. Imagine a curve drawn on a graph—integrals help us find the area under that curve, which is useful in physics, economics, and engineering.

Key Takeaway:

• Differentiation → Finds rate of change
• Integration → Finds original function 

Once you understand this relationship, the rest of the chapter becomes much easier to grasp.

Types of Integrals 

In calculus, integrals are mainly of two types:

1. Indefinite Integrals
2. Definite Integrals 

Understanding the difference between them is very important because they serve different purposes.

Indefinite Integrals 

Indefinite integrals are integrals without limits. They represent a general solution or a family of functions.

For example:

Why “family of functions”?

Because different values of give different functions:

All of these have the same derivative:

Examples:

Common Mistakes:

• Forgetting to add + C
• Applying wrong formulas
• Confusing with definite integrals 

Key Idea:

Indefinite integrals give a function, not a fixed answer.

Definite Integrals 

Definite integrals have limits, so they give a fixed numerical value.

For example:

Here:

• 0 → Lower limit
• 2 → Upper limit 

What do limits mean?

Limits tell us the interval over which we are calculating the result—usually the area under a curve.

Solve Step-by-Step:

First integrate:

Now apply limits:

Key Idea:

Definite integrals give a number, not a function.

Difference Explained (Simple Way)

• Indefinite integral → Like a general formula (many answers)
• Definite integral → Like a final value (one answer) 

Or simply:

• No limits → Add + C
• With limits → No + C, gives fixed value 

Important Terms in Integration 

Before solving integrals, it’s important to understand the basic terms used in integration.

Explanation:

Each part of an integral has a specific role. The integrand is what you need to integrate, while dx tells you which variable you are working with. The constant C ensures that all possible answers are included.

Example Breakdown:

Consider:

• Integrand →
• Variable →
• dx → indicates integration with respect to

Solution:

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How to Identify Each Part in a Question:

Whenever you see an integral:

• Look inside ∫ → that’s your integrand
• Look at dx → tells you the variable
• Check if limits are present → tells you type of integral 

Key Tip:

Always read the integral carefully before solving—it helps you decide the right method and formula.

Standard Formulas of Integrals 

These formulas are the foundation of integration. Most questions are solved directly or indirectly using them.

Algebraic Functions 

The most important formula:

Why ?

Because when , the formula becomes:

which is undefined. That’s why it becomes a special case:

Examples:

Tip:

Just increase power by 1 and divide by new power.

Trigonometric Functions 

These are based on reverse differentiation.

Key Formulas:

Pattern Understanding:

• Derivative of sin → cos → so integral of cos → sin
• Derivative of cos → -sin → so integral of sin → -cos 

Examples:

Tip:

Think: “reverse derivative.”

Exponential Functions 

Formulas:

Difference between and :

• → simplest form (no denominator)
• → includes in denominator 

Examples:

Tip:

If base ≠ e → divide by the log of base

Logarithmic Function 

Formula:

Why special?

Because the power rule fails at , so we use a logarithmic form instead.

Examples:

Memory Tricks (Important!)

• Power rule → “Increase power, divide”
• sin → -cos
• cos → sin
• → same
• → log 

Learning patterns is easier than memorizing everything!

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Properties of Integrals 

Properties help simplify integrals and make calculations easier.

Linearity Property

Why it works:

Integration distributes over addition.

Example:

Constant Multiple Rule

Why it works:

Constants do not affect integration.

Example:

Reverse of Differentiation

Why it works:

Integration and differentiation cancel each other.

Example:

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When to Use These Properties:

• Break large expressions → use linearity
• Remove constants → use constant rule
• Check answers → use reverse property 

Common Mistakes:

• Forgetting to apply the property to all terms
• Missing constant C
• Incorrect splitting of terms 

Methods of Integration 

This is the most important section, as it teaches how to solve complex integrals.

Integration by Substitution 

When to Use:

• Function inside another function
• Expression like

Steps:

1. Assume a new variable
2. Differentiate it
3. Replace in integral
4. Solve and substitute back 

Example 1:

Let ,

Final Answer:

Example 2:

Let

Integration by Parts 

Formula:

ILATE Rule:

Choose based on priority:
I > L > A > T > E

Example 1:

Example 2:

Trigonometric Identities 

When to Use:

• Products like sin×cos
• Powers like sin², cos² 

Example 1:

Use identity:

Answer:

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Example 2:

Use:

Partial Fractions 

When to Use:

• Rational functions (fractions)
• Denominator can be factorized 

Steps:

1. Factor denominator
2. Split into simpler fractions
3. Integrate each part 

Example:

How to Identify Which Method to Use:

Final Tip:

The more you practice, the faster you’ll recognize the method. Initially, focus on pattern recognition rather than speed.

Important Integrals (Special Forms)

Certain integrals appear frequently in exams and are considered special forms because they don’t directly follow basic algebraic or trigonometric formulas.

Key Formulas:

• ∫ dx / (x² + a²) = (1/a) tan⁻¹(x/a) + C
• ∫ dx / √(a² − x²) = sin⁻¹(x/a) + C
• ∫ dx / (x² − a²) = (1/2a) ln |(x − a)/(x + a)| + C
• ∫ eˣ sinx dx = (eˣ/2)(sinx − cosx) + C
• ∫ eˣ cosx dx = (eˣ/2)(sinx + cosx) + C 

Where Are These Used?

These integrals are commonly used in:

• Trigonometric substitutions
• Definite integrals involving inverse functions
• Applications in physics (waves, motion) 

Example:

Evaluate ∫ dx / (x² + 4)

Solution:
Here, a² = 4 ⇒ a = 2
Using formula:
∫ dx / (x² + a²) = (1/a) tan⁻¹(x/a) + C

= (1/2) tan⁻¹(x/2) + C

These special forms are often hidden inside complex expressions, so recognizing them quickly is key.

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Solved Examples

Let’s solve a variety of integrals step-by-step:

Basic Level:

1. ∫ x² dx
= x³/3 + C

2. ∫ cosx dx
= sinx + C

Moderate Level:

3. ∫ (2x + 3) dx
Split the integral:
= ∫ 2x dx + ∫ 3 dx
= x² + 3x + C

4. ∫ x cosx dx
Use integration by parts:
Let u = x, dv = cosx dx
Then du = dx, v = sinx

= x sinx − ∫ sinx dx
= x sinx + cosx + C

Application-Based:

5. ∫ dx / (x² + 9)
= (1/3) tan⁻¹(x/3) + C

6. ∫ eˣ cosx dx
= (eˣ/2)(sinx + cosx) + C

7. ∫ (1/x) dx
= ln|x| + C

8. ∫ (x² + 1)/x dx
Split:
= ∫ x dx + ∫ (1/x) dx
= x²/2 + ln|x| + C

 Notice how breaking the integral into smaller parts simplifies the solution.

Quick Revision Summary (Cheat Sheet)

Here’s a quick cheat sheet to revise integration concepts faster before exams:

Key Formulas:

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
• ∫ (1/x) dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ sinx dx = −cosx + C
• ∫ cosx dx = sinx + C
• ∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
• ∫ dx/√(a² − x²) = sin⁻¹(x/a) + C 

Method Selection Shortcuts:

• Substitution → When you see a function inside another (like (x²+1), √(x+3))
• Integration by Parts → Product of functions (x·eˣ, x·sinx)
• Trigonometric Identities → Powers of sin, cos (sin²x, cos³x)
• Partial Fractions → Rational functions (polynomial/polynomial) 

1-Line Reminders:

• Always add + C in indefinite integrals
• Simplify the expression before solving
• Check if the integrand matches a known pattern
• Break complex integrals into smaller parts
• Use identities to reduce powers in trigonometry
• Revise formulas daily instead of cramming 

Pro Tip: If you’re stuck, try differentiating your answer to verify it.
This quick revision block can save valuable time and boost accuracy in exams!

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10. Practice Questions (Category-Wise)

Difficulty Progression:

Start with basic formula-based questions, move to multi-step problems, and finally attempt application-level integrals.

A. Basic Integrals

(Use direct formulas)

1. ∫ x³ dx
2. ∫ 5 dx
3. ∫ sinx dx
4. ∫ cosx dx
5. ∫ eˣ dx
6. ∫ (1/x) dx
7. ∫ 7x dx
8. ∫ x⁵ dx
9. ∫ sec²x dx
10. ∫ csc²x dx 

B. Substitution Method

(Use substitution to simplify expressions)

1. ∫ 2x (x² + 1)² dx
2. ∫ (3x² + 2x) dx
3. ∫ x√(x² + 4) dx
4. ∫ (1/(x + 1)) dx
5. ∫ (2x)/(x² + 5) dx
6. ∫ e^(2x) dx
7. ∫ cos(3x) dx
8. ∫ sin(5x) dx
9. ∫ x/(x² + 1) dx
10. ∫ (x² + 1)³ dx 

C. Integration by Parts

(Apply ILATE rule)

1. ∫ x eˣ dx
2. ∫ x sinx dx
3. ∫ x cosx dx
4. ∫ ln x dx
5. ∫ x² eˣ dx
6. ∫ x² sinx dx
7. ∫ x ln x dx
8. ∫ eˣ cosx dx
9. ∫ eˣ sinx dx
10. ∫ x³ cosx dx 

D. Trigonometric Integrals

(Use identities)

1. ∫ sin²x dx
2. ∫ cos²x dx
3. ∫ sinx cosx dx
4. ∫ tanx dx
5. ∫ secx tanx dx
6. ∫ cosecx cotx dx
7. ∫ sin³x dx
8. ∫ cos³x dx
9. ∫ sec²x dx
10. ∫ cot²x dx 

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E. Partial Fractions

(Break rational expressions)

1. ∫ 1/(x² − 1) dx
2. ∫ (2x + 3)/(x² + x − 2) dx
3. ∫ 1/(x(x + 1)) dx
4. ∫ (x + 2)/(x² + 3x + 2) dx
5. ∫ 1/(x² + 5x + 6) dx
6. ∫ (3x)/(x² − 4) dx
7. ∫ (x² + 1)/(x³ − x) dx
8. ∫ 1/(x(x − 2)) dx
9. ∫ (2x + 1)/(x² + x) dx
10. ∫ 1/(x² − 9) dx 

Most Expected Exam Questions

When it comes to exams, certain integration problems appear repeatedly because they test core concepts and method selection skills. Practicing these patterns can significantly boost your confidence and accuracy.

Important Question Patterns:

1. ∫ x eˣ dx
Tests integration by parts (product of algebraic and exponential functions).

2. ∫ dx / (x² + a²)
A standard inverse trigonometric form leading to tan⁻¹(x/a).

3. ∫ sin²x dx
Requires trigonometric identity:
sin²x = (1 − cos2x)/2

4. ∫ x sinx dx
Again uses integration by parts, checking your understanding of the ILATE rule.

5. ∫ dx / (x² − a²)
Solved using partial fractions, a very common exam favorite.

6. ∫ eˣ cosx dx
A classic repeated integration by parts problem with a standard result.

7. ∫ (2x)/(x² + 1) dx
Tests the substitution method, where the denominator derivative is present in the numerator.

Why These Matter:

These questions are not random—they are designed to check:

• Your ability to identify the right method
• Your knowledge of standard formulas
• Your problem-solving speed 

Pro Tip: If you master these patterns, you’ll be able to solve a large portion of exam questions quickly and accurately.

Tips to Master Integrals

Mastering integration is all about pattern recognition + practice.

How to Remember Formulas:

• Link integrals to derivatives (reverse thinking)
• Group formulas: algebraic, trig, exponential
• Revise daily using flashcards 

How to Choose Method:

• If the derivative of the inner function exists → Substitution
• Product of functions → Integration by Parts
• Trig powers → Use identities
• Rational function → Partial Fractions 

Exam Tips:

• Solve easy questions first
• Don’t get stuck on one problem
• Always check for simplification before solving 

Time-Saving Tricks:

• Split complex integrals into smaller parts
• Memorize common results
• Avoid unnecessary steps 

Consistency beats intelligence in integration.

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Common Mistakes Students Make

Avoid these common errors to improve accuracy:

Frequent Mistakes:

• Forgetting + C in indefinite integrals
• Applying wrong formula
• Ignoring absolute value in logarithms
• Making sign errors in trigonometric integrals
• Wrong substitution choice
• Skipping simplification before solving
• Mixing up identities
• Incorrect partial fraction decomposition 

Why These Matter:

Even a small mistake can lead to a completely wrong answer in integration.

Pro Tip:

After solving:

• Differentiate your answer to verify
• Recheck signs and constants 

Accuracy + clarity = full marks.

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Mastering Integration Step-by-Step

Integration may seem difficult at first, but once you understand the key patterns, formulas, and methods, it becomes much easier. This guide covered everything from basics to advanced techniques like substitution and integration by parts.

Focus on understanding, not memorizing, and practice consistently. Integration is widely used in fields like physics, engineering, and economics, making it an important skill to master. Start small, revise regularly, and keep practicing—confidence will follow!

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