Integration is undoubtedly a very important chapter in Class 12 Maths. At first, many students find it quite difficult. In this blog, you will not only get the complete list of Class 12 integration all formulas but also find them explained in the simplest and easiest language possible. However, once you clearly understand the basic concepts and formulas, everything becomes much easier to solve.
In this blog, you will not only get the complete list of Class 12 integration all formulas but also find them explained in the simplest and easiest language possible.
Moreover, we have also included short tricks and smart tips to help you remember these Class 12 integration all formulas quickly and effectively. As a result, your preparation will become faster and more confident.
So, instead of getting confused, let’s go step by step. That way, you will learn all the important Class 12 integration all formulas in a clear and structured manner.
Eventually, with regular revision, you will master this chapter easily.
What Is Integration?
To begin with, let us first understand what integration means. Simply put, integration is the reverse of differentiation. While differentiation breaks things down, integration puts them together.
For example, if the derivative of x² is 2x, then the integration of 2x is x² (plus a constant).
In Class 12, integration helps to:
• Find the area under curves
• Calculate total distance
• Reconstruct functions from their derivatives
Types of Integration in Class 12
In your syllabus, you will find different types of integration methods. They include:
- Indefinite Integration
- Definite Integration
- Integration by Substitution
- Integration by Parts
- Integration using Partial Fractions
Now, let’s look at the Class 12 integration all formulas one by one.
Class 12 Integration All Formulas – Indefinite Type
Indefinite integration means solving without any upper or lower limits. Below are the most common formulas.
🔹 Basic Integration Formulas
∫1 dx = x + C
∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (where n ≠ -1)
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫aˣ dx = aˣ/ln a + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec²x dx = tan x + C
∫cosec²x dx = -cot x + C
∫sec x·tan x dx = sec x + C
∫cosec x·cot x dx = -cosec x + C
Quick Tip: If you remember the derivative of a function, then its integration is the reverse.
🔹 Trigonometric Integration Formulas
Some more formulas you must know:
- ∫tan x dx = ln|sec x| + C
- ∫cot x dx = ln|sin x| + C
- ∫sec x dx = ln|sec x + tan x| + C
- ∫cosec x dx = ln|cosec x – cot x| + C
Class 12 Integration All Formulas – By Substitution
When direct integration is not possible, we can use substitution. This method makes solving easier.
🔹 Standard Substitution Formulas
∫1/√(a² – x²) dx = sin⁻¹(x/a) + C
∫1/√(a² + x²) dx = ln|x + √(x² + a²)| + C
∫1/√(x² – a²) dx = ln|x + √(x² – a²)| + C
∫dx/(a² + x²) = (1/a) tan⁻¹(x/a) + C
∫dx/(x² – a²) = (1/2a) ln|(x – a)/(x + a)| + C
Shortcut:
• Use x = a sinθ for √(a² – x²)
• Use x = a tanθ for √(a² + x²)
• Use x = a secθ for √(x² – a²)
Class 12 Integration All Formulas – By Parts
When two functions are multiplied and you cannot use basic formulas, apply this method.
Formula of Integration by Parts
∫u·v dx = u∫v dx – ∫(du/dx)·∫v dx
To choose the correct function for u, use the ILATE rule:
• I – Inverse Trigonometric
• L – Logarithmic
• A – Algebraic
• T – Trigonometric
• E – Exponential
Pick the one that appears first in ILATE as u.
Class 12 Integration All Formulas – Partial Fractions
This is useful when you are given a rational expression (one polynomial divided by another). You can split it into simple parts and integrate them one by one.
🔹 Example Formula:
∫1/(x² – 1) dx
= ∫[1/2(1/(x – 1)) – 1/2(1/(x + 1))] dx
= (1/2) ln|x – 1| – (1/2) ln|x + 1| + C
Pro Tip: Always factor the denominator before splitting.
Definite Integration Formulas
Now let’s talk about definite integration. Here, the result is a fixed value, not a function.
🔹 Definite Integration Properties
- ∫ₐᵇ f(x) dx = F(b) – F(a)
- ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx
- ∫ₐᵃ f(x) dx = 0
- ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a – x) dx
- If f(a – x) = f(x), then ∫₀ᵃ f(x) dx = 2∫₀ᵃ⁄² f(x) dx
- If f(a – x) = -f(x), then ∫₀ᵃ f(x) dx = 0
Remember: Use properties when the limits are from 0 to a or symmetric.
Class 12 Integration All Formulas: Summary Chart
How to Remember Class 12 Integration All Formulas Easily
Here are a few easy ways to remember everything:
- Make a Formula Sheet – Keep it with you for daily revision.
- Group Formulas – Divide into categories: basic, trigonometric, etc.
- Practice Every Day – Solve at least 5 problems daily.
- Use Flashcards – Write one formula per card and review regularly.
- Say It Out Loud – Speaking while learning helps you remember better.
Practice Examples Using Formulas
Here are a few questions and solutions:
🔹 Example 1
Find ∫x³ dx
= x⁴/4 + C
🔹 Example 2
Find ∫1/(1 + x²) dx
= tan⁻¹x + C
🔹 Example 3
Find ∫x·eˣ dx
Use integration by parts
= x·eˣ – ∫eˣ dx = x·eˣ – eˣ + C
Final Tips for Exam Success
score high marks in integration:
First of all, identify the type of integration question carefully.
Next, choose and apply the correct integration formula accordingly.
After that, write down all steps clearly and neatly to avoid confusion.
Moreover, don’t forget to add +C in case of indefinite integration.
On the other hand, for definite integration, apply the limits properly.
In addition, double-check your solution to catch any small errors.
Finally, practice regularly to build speed, accuracy, and confidence.
Conclusion
So now, you finally have the complete list of Class 12 integration formulas along with short tricks, clearly explained in simple and easy language. First of all, don’t worry if it seems tough at the beginning. After all, every new topic feels tricky at first. However, as you keep going, things will slowly start making sense.
Moreover, with regular practice and frequent revision, you will definitely understand the concepts better. In fact, the more you revise, the easier it becomes. Therefore, make it a habit to solve a few questions daily. Not only will this boost your confidence, but it will also improve your speed. Eventually, you will see great results.
So, keep learning, stay confident, and above all, don’t forget – practice is the key! In addition, consistent effort will lead to mastery. Furthermore, by breaking down complex problems step-by-step, you will find them less intimidating. Consequently, your problem-solving skills will sharpen over time. As a result, your exam preparation will become much more effective.
To sum up
Patience and persistence are your best allies in this journey.So now, you finally have the complete list of Class 12 integration formulas along with short tricks, clearly explained in simple and easy language. First of all, don’t worry if it seems tough at the beginning. After all, every new topic feels tricky at first.
However, as you keep going, things will slowly start making sense. Moreover, with regular practice and frequent revision, you will definitely understand the concepts better. In fact, the more you revise, the easier it becomes. Therefore, make it a habit to solve a few questions daily. Not only will this boost your confidence, but it will also improve your speed.
Eventually, you will see great results. So, keep learning, stay confident, and above all, don’t forget – practice is the key! In addition, consistent effort will lead to mastery.
Furthermore, by breaking down complex problems step-by-step, you will find them less intimidating. Consequently, your problem-solving skills will sharpen over time. As a result, your exam preparation will become much more effective. To sum up, patience and persistence are your best allies in this journey.