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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
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