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Important Indefinite Integral Formulas for JEE and NEET
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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
= –
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2
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n–2
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I
n
= –
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+
3
a ) 1 n (
2
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I
n–2
? I
n
=
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) x a ( x
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+
) 2 n (
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2
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I
n–2
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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.
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