Indefinite Integration Notes for IIT JEE, Download PDF!

By Subrato Banerjee|Updated : December 13th, 2018

Indefinite integration is one of the most important topics for preparation of any engineering entrance examination. Thus, it is necessary for every candidate to be well versed with the formulas and concepts of indefinite integration. Revise the notes and attempt more and more questions on this topic. Every year 3-4 questions are asked in JEE Main/ JEE Advanced. Do not forget to download Indefinite Integration notes PDF from the end of the post.


1. Integral of a function, integrand

If the derivative of a function f(x) is F(x) i.e. if


we say that f(x) is an integral or primitive of F(x) and in symbols, we write 


The letter x in dx denotes that the integration is to be performed with respect to the variable x.

2. Integral as an anti-derivative

The process of determining an integral of a function is called integration and the function to be integrated is called the integrand.

Since integration and differentiation are inverse processes we have


3. Indefinite Integration, General Integral, Arbitrary Constant 

If f(x) is an integral of F(x) then f(x)+c is also an integral of F(x), c being a constant what so ever for





where c is an arbitrary constant. If f(x) be an integral of F(x), then f(x)+c is its general integral.

As c is indefinite in nature, that’s why such integration is known as indefinite integration.

4. Theorems on Indefinite Integration

∫af(x)dx=a∫f(x)dx  where a is a constant

∫(f(x)±g(x))dx=∫f(x)dx ± ∫g(x)dx

5. Fundamental Integrals


6. Integration by substitution

In this method the integral ∫f(x)dx is expressed in terms of another integral where some other variables say t is the independent variable; x and t being connected by some suitable relation x=g(t) . It leads to the result  ∫f(x)dx =∫f(g(t)). g' (t) dt

7. Standard Substitutions

(a). For terms of the form x2 +a2 or (x2 +a)1/2 put x=a tan t

(b). For terms of the form x2 - a2 or (x2 - a)1/2 put x=a sec t

(c). For terms of the form  a2 - x2 or (a2 - x2 )1/2 put x=a sin t

(d). When both (a + x )1/2 (a - x )1/2 are present put x=a cos 2t

(e). For the terms of the form ((x-a)(b-x))1/2 put x=a cost + b sint

(f). For terms of the form (x±(x2 - a)1/2 )n put t=x±(x2 - a)1/2

8. Integration by parts

If u,v be two differential functions of x. We have


Integrating both sides we get,


 Let u=f(x) and 






So, ∫f(x) g(x) dx = f(x) ∫g(x)dx - ∫ [(∫g(x)dx ). f' (x)] dx

In words, this states that,

The integral of (product of two functions) =first function*integral of the second-integral of (derivative of first function*integral of second function)

The first and second function is decided by the ILATE rule where






9. Integration by Partial Fractions

If the integral is in the form of an algebraic fraction which cannot be integrated then the fraction needs to be decomposed into partial fractions.

Rules for expressing in partial fraction:

1. The numerator must be at least one degree less than the denominator.

2. For every factor (ax+b) in the denominator, there is a partial fraction 12

3. If a factor is repeated in the denominator n times then that partial fraction should be written n times with degree 1 through n

4. For a factor of the form ( ax2 + bx +c )in the denominator, there will be a partial fraction of the form


10. Integration using trigonometric functions

Integrations of the form ∫sinmx cosnx can be solved for the following cases
 can be solved for the following cases

Case 1: Either m or n is odd

Write all even powers of sin x in terms of cos x

For the remaining sin x write sinx dx as - d(cos x)

Case 2: When both m and n are even

Use double angle formula


Case 3: When m and n are not all positive integers

Convert sin x and cos x in terms of sec x and tan x

(i). If the power of sec x is even, use substitution u=tan x ⇒ du=sec2x dx

Use 1+tan2x = sec2x

(ii). If the power of tan x is negative odd, use substitution with u = sec x, du = sec x tan x dx, and convert remaining powers of tan to a function of u using tan2x =  sec2x - 1 . This works if m ≥ 1.

(iii). If the power of sec x is odd and tan x is even, use 1+tan2x = sec2x and convert all sec x into tan x


11. Special Integrals


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Satyam Kumar

Satyam KumarDec 13, 2018

Provide notes of rotational mechanics plese
Anwitha Anakarla
Thanks .. please provide inorganic chemistry qualitative analysis short notes
Saurabh Pawar
Sir provide mathematical resoning pdf
Vikas Yadav
Can u send me notes of definite integration
Sundaram Singh
thanks sir your quizis very IMP for me
Yash Sanap

Yash SanapAug 10, 2019

LIATE rule correct,you are incorrectly formula
Bammidi Satyam
Please provide coordination and compounds notes

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