Integration Formula for Class 12: A Complete Guide

In this blog post, you will understand the concept of Integration Formula for Class 12 as Integration is one of the most important and fascinating concepts in Class 12 Mathematics.

It is essentially the reverse process of differentiation and is used in a wide range of applications, from finding areas under curves to solving complex equations in physics and economics. In this blog post, we’ll dive into the essential integration formulas that every Class 12 student should master. If you’re preparing for board exams or competitive exams like JEE, these formulas are key.

Integration Formula for Class 12: What is Integration?

In simple terms, integration is the process of finding the integral or antiderivative of a function. If differentiation helps in finding the rate of change, integration helps in finding the total accumulation of quantities. For example, if you differentiate the position of an object, you get its velocity. To reverse the process, you integrate the velocity to find the position.

There are two main types of integrals:

1. Indefinite Integrals – These represent a family of functions and include an arbitrary constant (C).
2. Definite Integrals – These calculate the exact area under the curve between two limits.

Basic Integration Formulas: Integration Formula for Class 12

Here’s a list of essential integration formulas that are part of the Class 12 syllabus:

1. ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C (where n ≠ -1)
   • This is the most basic formula for integrating power functions.
   • Example: ∫ x² dx = (x³ / 3) + C
2. ∫ eˣ dx = eˣ + C
   • The integral of the exponential function is itself.
   • Example: ∫ e²ˣ dx = (e²ˣ / 2) + C
3. ∫ 1/x dx = ln|x| + C
   • The natural logarithm is the antiderivative of 1/x.
   • Example: ∫ 1/(2x) dx = (1/2) ln|x| + C
4. ∫ sin(x) dx = -cos(x) + C
   • Integration of sine gives the negative cosine.
   • Example: ∫ sin(2x) dx = (-cos(2x) / 2) + C
5. ∫ cos(x) dx = sin(x) + C
   • The antiderivative of cosine is sine.
   • Example: ∫ cos(3x) dx = (sin(3x) / 3) + C
6. ∫ sec²(x) dx = tan(x) + C
   • The integral of sec²(x) results in tan(x).
   • Example: ∫ sec²(2x) dx = (tan(2x) / 2) + C
7. ∫ 1/(x² + a²) dx = (1/a)tan^-1(x/a) + C
   • This formula is used when dealing with expressions involving quadratic terms.
   • Example: ∫ 1/(x² + 4) dx = (1/2)tan^-1(x/2) + C

Integration Using Substitution: Integration Formula for Class 12

One of the most common methods used in solving integrals is substitution. In this method, we substitute a part of the function with a new variable, simplify the integral, and then revert to the original variable after integration.

Example:
∫ (2x)/(x² + 1) dx
Here, we use substitution:
Let u = x² + 1, then du/dx = 2x or du = 2x dx.
Now, the integral becomes:
∫ 1/u du = ln|u| + C
Substitute u = x² + 1 back:
Result: ln|x² + 1| + C

Integration by Parts: Integration Formula for Class 12

Another key method to integrate functions is integration by parts, based on the product rule of differentiation.

The integration of (First x Second) = (First) x (Integration of Second)

                                             – Integration of {Diff of First x Integration of 2nd}

For this Method, we have to follow the priority order of the functions given below-

ILATE Rule-

Inverse trigonometric (I)

Logarithmic (L)

Algebraic (A)

Trigonometric (T)

Exponential (E)

Example-

If we have to find  

Then, we will take x (Algebraic Function) as first function and sinx (Trigonometric Function) as the second function.                       

Finally, the above question can be solved as follows-

Formulas for Some Special or Particular Integrals: Integration Formula for Class 12

1.

2.

3.

4.

5.

6.

7.

8.       

9.

   

Conclusion

Integration is an essential chapter for Class 12 students, and mastering the formulas and techniques will help you not only in your board exams but also in competitive exams like JEE. By practicing the basic formulas and learning methods like substitution and integration by parts, you will develop a strong foundation in calculus.

Keep practicing, and soon, integration will become one of your strongest areas in mathematics.

FAQs-

1. What is Integration in Class 12 Mathematics?

Answer:
Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. In Class 12 Mathematics, students learn various techniques of integration, such as integration by substitution, integration by parts, and the use of standard formulas.

2.How is Integration by Substitution Done?

Answer:
Integration by substitution is a technique used to simplify integrals by substituting a part of the expression with a new variable. This method works well when the integral contains a function and its derivative. The general process involves substituting u=f(x), then finding du, and rewriting the integral in terms of u.

3.What is the Application of Definite Integrals in Class 12?

Answer:
Definite integrals are used to find the area under a curve between two points. In Class 12, students learn how to calculate the exact area between a function and the x-axis over a given interval [a,b] using definite integrals. The result is a numerical value, unlike indefinite integrals that include a constant of integration C.

$C$

4.What are Some Common Mistakes to Avoid in Integration Problems?

Common mistakes in integration include:

• Forgetting the constant of integration CCC in indefinite integrals.
• Incorrectly applying formulas or not simplifying the expression properly before integrating.
• Confusing differentiation rules with integration rules.
• Misusing the limits in definite integrals.

5.How Important is Integration for Class 12 Exams?

Answer:
Integration is a crucial topic in Class 12 Mathematics and is frequently tested in both board exams and competitive exams like JEE. Mastering the formulas, techniques, and applications of integration is essential for scoring well.

6.What is the Difference Between Definite and Indefinite Integrals?

Answer:

• Indefinite Integrals: These represent a family of functions and include a constant of integration C. They provide the general form of the antiderivative of a function.
• Definite Integrals: These give the exact area under the curve between two specific limits and result in a numerical value, without a constant C.

.7. Are There Any Tips to Master Integration for Class 12?

Answer:
Yes, some tips include:

• Practice regularly to familiarize yourself with various types of integrals.
• Memorize key integration formulas.
• Understand the logic behind each method (e.g., substitution, parts) rather than just applying formulas blindly.
• Work on previous year’s exam papers and sample questions to improve speed and accuracy.

Also Read-

https://pcmconceptclear.com/integration-using-trigonometric-identities/

https://pcmconceptclear.com/integration-by-substitution-maths12/

https://pcmconceptclear.com/integration-basic-concept-and-formula/