JEE Exam > JEE Notes > Mathematics (Maths) Main & Advanced > Important Indefinite Integral Formulas for JEE and NEET
Important Indefinite Integral Formulas for JEE and NEET
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Page # 14
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g ?(x) = f(x) then,
?
f(x)
dx = g(x)
+ c ? ?
d
dx
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i)
?
(ax + b)
n
dx =
? ?
? ?
ax b
a n
n
?
?
?1
1
+ c, n ? ?1
(ii)
?
dx
ax b ?
=
1
a
?n (ax + b) + c
(iii)
?
e
ax+b
dx =
1
a
e
ax+b
+ c
(iv)
?
a
px+q
dx =
1
p
a
na
px q ?
?
+ c; a > 0
(v)
?
sin (ax
+ b) dx = ?
1
a
cos (ax
+ b)
+ c
(vi)
?
cos
(ax
+ b) dx =
1
a
sin (ax
+ b) + c
(vii)
?
tan(ax
+ b) dx =
1
a
?n sec
(ax
+ b)
+ c
(viii)
?
cot(ax
+
b)
dx =
1
a
?n sin(ax +
b)+ c
(ix)
?
sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x)
?
cosec²(ax + b) dx =
?
1
a
cot(ax + b)+ c
Page 2
Page # 14
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g ?(x) = f(x) then,
?
f(x)
dx = g(x)
+ c ? ?
d
dx
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i)
?
(ax + b)
n
dx =
? ?
? ?
ax b
a n
n
?
?
?1
1
+ c, n ? ?1
(ii)
?
dx
ax b ?
=
1
a
?n (ax + b) + c
(iii)
?
e
ax+b
dx =
1
a
e
ax+b
+ c
(iv)
?
a
px+q
dx =
1
p
a
na
px q ?
?
+ c; a > 0
(v)
?
sin (ax
+ b) dx = ?
1
a
cos (ax
+ b)
+ c
(vi)
?
cos
(ax
+ b) dx =
1
a
sin (ax
+ b) + c
(vii)
?
tan(ax
+ b) dx =
1
a
?n sec
(ax
+ b)
+ c
(viii)
?
cot(ax
+
b)
dx =
1
a
?n sin(ax +
b)+ c
(ix)
?
sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x)
?
cosec²(ax + b) dx =
?
1
a
cot(ax + b)+ c
Page # 15
(xi)
?
secx dx = ?n (secx + tanx) + c OR ?n tan
?
4 2
?
?
?
?
?
?
?
x
+ c
(xii)
?
cosec x dx = ?n (cosecx ? cotx) + c
OR ?n tan
x
2
+ c OR ? ?n (cosecx + cotx) + c
(xiii)
?
d x
a x
2 2
?
= sin
?1
x
a
+ c
(xiv)
?
d x
a x
2 2
?
=
1
a
tan
?1
x
a
+ c
(xv)
?
2 2
a x | x |
x d
?
=
1
a
sec
?1
x
a
+ c
(xvi)
?
?
2 2
a x
x d
= ?n
? ?
x x a ? ?
2 2
+ c
(xvii)
?
d x
x a
2 2
?
= ?n
? ?
x x a ? ?
2 2
+ c
(xviii)
?
d x
a x
2 2
?
=
1
2a
?n
x a
x a
?
?
+ c
(xix)
?
d x
x a
2 2
?
=
1
2a
?n
a x
a x
?
?
+ c
(xx)
?
a x
2 2
? dx =
x
2
a x
2 2
? +
a
2
2
sin
?1
x
a
+ c
(xxi)
?
x a
2 2
? dx =
x
2
x a
2 2
? +
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
(xxii)
?
x a
2 2
? dx =
x
2
x a
2 2
? ?
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
Page 3
Page # 14
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g ?(x) = f(x) then,
?
f(x)
dx = g(x)
+ c ? ?
d
dx
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i)
?
(ax + b)
n
dx =
? ?
? ?
ax b
a n
n
?
?
?1
1
+ c, n ? ?1
(ii)
?
dx
ax b ?
=
1
a
?n (ax + b) + c
(iii)
?
e
ax+b
dx =
1
a
e
ax+b
+ c
(iv)
?
a
px+q
dx =
1
p
a
na
px q ?
?
+ c; a > 0
(v)
?
sin (ax
+ b) dx = ?
1
a
cos (ax
+ b)
+ c
(vi)
?
cos
(ax
+ b) dx =
1
a
sin (ax
+ b) + c
(vii)
?
tan(ax
+ b) dx =
1
a
?n sec
(ax
+ b)
+ c
(viii)
?
cot(ax
+
b)
dx =
1
a
?n sin(ax +
b)+ c
(ix)
?
sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x)
?
cosec²(ax + b) dx =
?
1
a
cot(ax + b)+ c
Page # 15
(xi)
?
secx dx = ?n (secx + tanx) + c OR ?n tan
?
4 2
?
?
?
?
?
?
?
x
+ c
(xii)
?
cosec x dx = ?n (cosecx ? cotx) + c
OR ?n tan
x
2
+ c OR ? ?n (cosecx + cotx) + c
(xiii)
?
d x
a x
2 2
?
= sin
?1
x
a
+ c
(xiv)
?
d x
a x
2 2
?
=
1
a
tan
?1
x
a
+ c
(xv)
?
2 2
a x | x |
x d
?
=
1
a
sec
?1
x
a
+ c
(xvi)
?
?
2 2
a x
x d
= ?n
? ?
x x a ? ?
2 2
+ c
(xvii)
?
d x
x a
2 2
?
= ?n
? ?
x x a ? ?
2 2
+ c
(xviii)
?
d x
a x
2 2
?
=
1
2a
?n
x a
x a
?
?
+ c
(xix)
?
d x
x a
2 2
?
=
1
2a
?n
a x
a x
?
?
+ c
(xx)
?
a x
2 2
? dx =
x
2
a x
2 2
? +
a
2
2
sin
?1
x
a
+ c
(xxi)
?
x a
2 2
? dx =
x
2
x a
2 2
? +
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
(xxii)
?
x a
2 2
? dx =
x
2
x a
2 2
? ?
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
Page # 16
3.
If we subsitute f(x) = t, then f ?(x) dx = dt
4. Integration by Part :
? ?
?
) x ( g ) x ( f dx = f(x) ? ?
?
) x ( g dx – ? ? ? ? dx dx ) x ( g ) x ( f
dx
d
? ?
?
?
?
?
?
?
5. Integration of type
2
2
2
dx dx
, , ax bx c dx
ax bx c
ax bx c
? ?
? ?
? ?
? ? ?
Make the substitution
b
x t
2a
? ?
6. Integration of type
2
2
2
pxq pxq
dx, dx, (px q) ax bx c dx
ax bx c
ax bx c
? ?
? ? ?
? ?
? ?
? ? ?
Make the substitution x +
b
2a
= t , then split the integral as some of two
integrals one containing the linear term and the other containing constant
term.
7. Integration of trigonometric functions
(i) 2
dx
a bsin x ?
? OR
2
dx
a bcos x ?
?
OR 2 2
dx
asin x bsinx cosx ccos x ? ?
? put tan x = t.
(ii)
dx
a bsinx ?
? OR
dx
a bcos x ?
?
OR
dx
a bsinx c cos x ? ?
? put tan
x
2
= t
(iii)
a.cos x b.sin x c
.cosx m.sinx n
? ?
? ?
?
?
dx. Express Nr ? A(Dr) + B
d
dx
(Dr) + c & proceed.
Page 4
Page # 14
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g ?(x) = f(x) then,
?
f(x)
dx = g(x)
+ c ? ?
d
dx
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i)
?
(ax + b)
n
dx =
? ?
? ?
ax b
a n
n
?
?
?1
1
+ c, n ? ?1
(ii)
?
dx
ax b ?
=
1
a
?n (ax + b) + c
(iii)
?
e
ax+b
dx =
1
a
e
ax+b
+ c
(iv)
?
a
px+q
dx =
1
p
a
na
px q ?
?
+ c; a > 0
(v)
?
sin (ax
+ b) dx = ?
1
a
cos (ax
+ b)
+ c
(vi)
?
cos
(ax
+ b) dx =
1
a
sin (ax
+ b) + c
(vii)
?
tan(ax
+ b) dx =
1
a
?n sec
(ax
+ b)
+ c
(viii)
?
cot(ax
+
b)
dx =
1
a
?n sin(ax +
b)+ c
(ix)
?
sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x)
?
cosec²(ax + b) dx =
?
1
a
cot(ax + b)+ c
Page # 15
(xi)
?
secx dx = ?n (secx + tanx) + c OR ?n tan
?
4 2
?
?
?
?
?
?
?
x
+ c
(xii)
?
cosec x dx = ?n (cosecx ? cotx) + c
OR ?n tan
x
2
+ c OR ? ?n (cosecx + cotx) + c
(xiii)
?
d x
a x
2 2
?
= sin
?1
x
a
+ c
(xiv)
?
d x
a x
2 2
?
=
1
a
tan
?1
x
a
+ c
(xv)
?
2 2
a x | x |
x d
?
=
1
a
sec
?1
x
a
+ c
(xvi)
?
?
2 2
a x
x d
= ?n
? ?
x x a ? ?
2 2
+ c
(xvii)
?
d x
x a
2 2
?
= ?n
? ?
x x a ? ?
2 2
+ c
(xviii)
?
d x
a x
2 2
?
=
1
2a
?n
x a
x a
?
?
+ c
(xix)
?
d x
x a
2 2
?
=
1
2a
?n
a x
a x
?
?
+ c
(xx)
?
a x
2 2
? dx =
x
2
a x
2 2
? +
a
2
2
sin
?1
x
a
+ c
(xxi)
?
x a
2 2
? dx =
x
2
x a
2 2
? +
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
(xxii)
?
x a
2 2
? dx =
x
2
x a
2 2
? ?
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
Page # 16
3.
If we subsitute f(x) = t, then f ?(x) dx = dt
4. Integration by Part :
? ?
?
) x ( g ) x ( f dx = f(x) ? ?
?
) x ( g dx – ? ? ? ? dx dx ) x ( g ) x ( f
dx
d
? ?
?
?
?
?
?
?
5. Integration of type
2
2
2
dx dx
, , ax bx c dx
ax bx c
ax bx c
? ?
? ?
? ?
? ? ?
Make the substitution
b
x t
2a
? ?
6. Integration of type
2
2
2
pxq pxq
dx, dx, (px q) ax bx c dx
ax bx c
ax bx c
? ?
? ? ?
? ?
? ?
? ? ?
Make the substitution x +
b
2a
= t , then split the integral as some of two
integrals one containing the linear term and the other containing constant
term.
7. Integration of trigonometric functions
(i) 2
dx
a bsin x ?
? OR
2
dx
a bcos x ?
?
OR 2 2
dx
asin x bsinx cosx ccos x ? ?
? put tan x = t.
(ii)
dx
a bsinx ?
? OR
dx
a bcos x ?
?
OR
dx
a bsinx c cos x ? ?
? put tan
x
2
= t
(iii)
a.cos x b.sin x c
.cosx m.sinx n
? ?
? ?
?
?
dx. Express Nr ? A(Dr) + B
d
dx
(Dr) + c & proceed.
Page # 17
8.
2
4 2
x 1
dx
x Kx 1
?
? ?
?
where K is any constant.
Divide Nr & Dr by x² & put x ?
1
x
= t.
9. Integration of type
dx
(ax b) px q ? ?
? OR 2
dx
(ax bx c) px q ? ? ?
? ; put px + q = t
2
.
10. Integration of type
2
dx
(ax b) px qx r ? ? ?
? , put ax + b =
1
t
;
2 2
dx
(ax b) px q ? ?
? , put x =
1
t
Page 5
Page # 14
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g ?(x) = f(x) then,
?
f(x)
dx = g(x)
+ c ? ?
d
dx
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i)
?
(ax + b)
n
dx =
? ?
? ?
ax b
a n
n
?
?
?1
1
+ c, n ? ?1
(ii)
?
dx
ax b ?
=
1
a
?n (ax + b) + c
(iii)
?
e
ax+b
dx =
1
a
e
ax+b
+ c
(iv)
?
a
px+q
dx =
1
p
a
na
px q ?
?
+ c; a > 0
(v)
?
sin (ax
+ b) dx = ?
1
a
cos (ax
+ b)
+ c
(vi)
?
cos
(ax
+ b) dx =
1
a
sin (ax
+ b) + c
(vii)
?
tan(ax
+ b) dx =
1
a
?n sec
(ax
+ b)
+ c
(viii)
?
cot(ax
+
b)
dx =
1
a
?n sin(ax +
b)+ c
(ix)
?
sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x)
?
cosec²(ax + b) dx =
?
1
a
cot(ax + b)+ c
Page # 15
(xi)
?
secx dx = ?n (secx + tanx) + c OR ?n tan
?
4 2
?
?
?
?
?
?
?
x
+ c
(xii)
?
cosec x dx = ?n (cosecx ? cotx) + c
OR ?n tan
x
2
+ c OR ? ?n (cosecx + cotx) + c
(xiii)
?
d x
a x
2 2
?
= sin
?1
x
a
+ c
(xiv)
?
d x
a x
2 2
?
=
1
a
tan
?1
x
a
+ c
(xv)
?
2 2
a x | x |
x d
?
=
1
a
sec
?1
x
a
+ c
(xvi)
?
?
2 2
a x
x d
= ?n
? ?
x x a ? ?
2 2
+ c
(xvii)
?
d x
x a
2 2
?
= ?n
? ?
x x a ? ?
2 2
+ c
(xviii)
?
d x
a x
2 2
?
=
1
2a
?n
x a
x a
?
?
+ c
(xix)
?
d x
x a
2 2
?
=
1
2a
?n
a x
a x
?
?
+ c
(xx)
?
a x
2 2
? dx =
x
2
a x
2 2
? +
a
2
2
sin
?1
x
a
+ c
(xxi)
?
x a
2 2
? dx =
x
2
x a
2 2
? +
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
(xxii)
?
x a
2 2
? dx =
x
2
x a
2 2
? ?
a
2
2
?n
?
?
?
?
?
?
?
?
? ?
a
a x x
2 2
+ c
Page # 16
3.
If we subsitute f(x) = t, then f ?(x) dx = dt
4. Integration by Part :
? ?
?
) x ( g ) x ( f dx = f(x) ? ?
?
) x ( g dx – ? ? ? ? dx dx ) x ( g ) x ( f
dx
d
? ?
?
?
?
?
?
?
5. Integration of type
2
2
2
dx dx
, , ax bx c dx
ax bx c
ax bx c
? ?
? ?
? ?
? ? ?
Make the substitution
b
x t
2a
? ?
6. Integration of type
2
2
2
pxq pxq
dx, dx, (px q) ax bx c dx
ax bx c
ax bx c
? ?
? ? ?
? ?
? ?
? ? ?
Make the substitution x +
b
2a
= t , then split the integral as some of two
integrals one containing the linear term and the other containing constant
term.
7. Integration of trigonometric functions
(i) 2
dx
a bsin x ?
? OR
2
dx
a bcos x ?
?
OR 2 2
dx
asin x bsinx cosx ccos x ? ?
? put tan x = t.
(ii)
dx
a bsinx ?
? OR
dx
a bcos x ?
?
OR
dx
a bsinx c cos x ? ?
? put tan
x
2
= t
(iii)
a.cos x b.sin x c
.cosx m.sinx n
? ?
? ?
?
?
dx. Express Nr ? A(Dr) + B
d
dx
(Dr) + c & proceed.
Page # 17
8.
2
4 2
x 1
dx
x Kx 1
?
? ?
?
where K is any constant.
Divide Nr & Dr by x² & put x ?
1
x
= t.
9. Integration of type
dx
(ax b) px q ? ?
? OR 2
dx
(ax bx c) px q ? ? ?
? ; put px + q = t
2
.
10. Integration of type
2
dx
(ax b) px qx r ? ? ?
? , put ax + b =
1
t
;
2 2
dx
(ax b) px q ? ?
? , put x =
1
t
F. INTEGRATION BY REDUCTION FORMULAE
Ex.47 If I
n
=
?
?
2 2 n
x a x dx, prove that I
n
= –
) 2 n (
) x a ( x
2 / 3 2 2 1 n
?
?
?
+
) 2 n (
) 1 n (
?
?
a
2
I
n–2
.
Sol. I
n
=
?
?
2 2 n
x a x dx =
?
?
?
dx } x a x .{ x
2 2 1 n
Applying integration by parts we get
= x
n–1
.
2 2 3/2
(a x )
3
? ?
? ? ?
? ?
?
? ?
? ?
+
n 2
(n 1)x
?
?
?
.
2 2 3/2
(a x )
3
? ?
? ? ?
?
? ?
? ?
? ?
dx
= –
3
) x a ( x
2 / 3 2 2 1 n
?
?
+
3
) 1 n ( ?
?
?
?
) x a .( x
2 2 2 n
2 2
x a ?
dx
? I
n
= –
3
) x a ( x
2 / 3 2 2 1 n
?
?
+
3
a ) 1 n (
2
?
I
n–2
–
3
) 1 n ( ?
I
n
? I
n
+
3
) 1 n ( ?
I
n
= –
3
) x a ( x
2 / 3 2 2 1 n
?
?
+
3
a ) 1 n (
2
?
I
n–2
?
?
?
?
?
?
? ?
3
2 n
I
n
= –
3
) x a ( x
2 / 3 2 2 1 n
?
?
+
3
a ) 1 n (
2
?
I
n–2
? I
n
=
) 2 n (
) x a ( x
2 / 3 2 2 1 n
?
?
?
+
) 2 n (
a ) 1 n (
2
?
?
I
n–2
Read More173 videos|510 docs|154 tests |
About this Document
4.96/5 Rating
Mar 17, 2026 Last updated
Document Description: Important Indefinite Integral Formulas for JEE and NEET for JEE 2026 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for Important Indefinite Integral Formulas for JEE and NEET have been prepared according to the JEE exam syllabus. Information about Important Indefinite Integral Formulas for JEE and NEET covers topics like and Important Indefinite Integral Formulas for JEE and NEET Example, for JEE 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Indefinite Integral Formulas for JEE and NEET.
Introduction of Important Indefinite Integral Formulas for JEE and NEET in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & Important Indefinite Integral Formulas for JEE and NEET in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Important Indefinite Integral Formulas for JEE and NEET
Description
Full syllabus notes, lecture & questions for Important Indefinite Integral Formulas for JEE and NEET - JEE | Plus exercises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download
Information about Important Indefinite Integral Formulas for JEE and NEET
In this doc you can find the meaning of Important Indefinite Integral Formulas for JEE and NEET defined & explained in the simplest way possible. Besides explaining types of Important Indefinite Integral Formulas for JEE and NEET theory, EduRev gives you an ample number of questions to practice Important Indefinite Integral Formulas for JEE and NEET tests, examples and also practice JEE tests
173 videos|510 docs|154 tests |
Download as PDF
 | Explore Courses for JEE exam |  |
Related Searches
pdf , mock tests for examination, Important Indefinite Integral Formulas for JEE and NEET, Important questions, Important Indefinite Integral Formulas for JEE and NEET, practice quizzes, Previous Year Questions with Solutions, MCQs, Objective type Questions, Summary, Semester Notes, study material, ppt, Important Indefinite Integral Formulas for JEE and NEET, Exam, Viva Questions, past year papers, video lectures, Extra Questions, shortcuts and tricks, Sample Paper, Free;