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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
 = – 
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
 
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