Indefinite Integrals
Let f ( x ) be a function. Then, the collection of all its primitives is called
the indefinite integral (or anti-derivative) of f ( x ) and is denoted by
∫ f ( x )dx. Integration as an inverse process of differentiation.
d
If
dx
{ φ ( x )} = f ( x ), then ∫ f ( x ) dx = φ ( x ) + C , where C is called the
constant of integration or arbitrary constant.
Symbols f ( x ) → Integrand
f ( x )dx → Element of integration
∫ → Sign of integral
φ ( x ) → Anti-derivative or primitive or integral of function f ( x )
The process of finding functions whose derivative is given, is called
anti-differentiation or integration.
Note The derivative of function is unique but integral of a function is not unique.
Some Standard Integral Formulae
Derivatives Indefinite Integrals
n+1
d x xn + 1
= x ,n ≠ −1 ∫x dx = + C, n ≠ − 1
n
(i) n
dx n + 1 n+1
d 1 1
(ii)
dx
(log e x) =
x ∫ x dx = log e | x| + C
d x
(e ) = e x ∫e dx = e x + C
x
(iii)
dx
d ax ax
= a , a > 0, a ≠ 1 ∫ a dx = +C
x x
(iv)
dx log e a log e a
d
(v)
dx
(− cos x) = sin x ∫ sin x dx = − cos x + C
, Derivatives Indefinite Integrals
d
(vi)
dx
(sin x) = cos x ∫ cos x dx = sin x + C
d
(vii)
dx
(tan x) = sec2 x ∫ sec2 x dx = tan x + C
d
(viii)
dx
(− cot x) = cosec2 x ∫ cosec2 x dx = − cot x + C
d
(ix)
dx
(sec x) = sec x tan x ∫ sec x tan x dx = sec x + C
d
(x)
dx
(− cosec x) = cosec x cot x ∫ cosec x cot x dx = − cosec x + C
(xi)
d
(log sin x) = cot x ∫ cot x dx = log|sin x| + C
dx = − log|cosec x| + C
(xii)
d
(− log cos x) = tan x ∫ tan x dx = − log|cos x| + C
dx
= log|sec x|+ C
(xiii)
d
dx
[log (sec x + tan x)] = sec x ∫ sec x dx = log|sec x + tan x| + C
π x
= log tan + + C
4 2
(xiv)
d
dx
[log (cosec x − cot x)] ∫ cosec x dx = log = cosec x
x
|cosec x − cot x| + C = log tan +C
2
sin− 1 = dx = sin−1 + C
d x 1 1 x
(xv) dx a a −x
2 2 ∫ a −x
2 2 a
−1 −1
cos −1 = ∫ dx = cos −1 + C
d x x
(xvi) dx a a −x
2 2 ∫ a −x
2 2 a
d 1 tan−1 x 1 1 1
tan−1 + C
x
(xvii) = 2 ∫ a2 dx =
dx a a a + x2 + x2 a a
1 cot −1 x −1 −1
cot −1 + C
d 1 x
(xviii) = ∫ a2 dx =
dx a a a2 + x2 + x2 a a
d 1 −1 x 1
sec −1 + C
1 1 x
sec = ∫x dx =
(xix) dx a a x x2 − a2 x −a
2 2 a a
d 1 cosec −1 x −1 −1
cosec −1 + C
1 x
= ∫x dx =
(xx) dx a a x x2 − a2 a
x −a
2 2 a
Let f ( x ) be a function. Then, the collection of all its primitives is called
the indefinite integral (or anti-derivative) of f ( x ) and is denoted by
∫ f ( x )dx. Integration as an inverse process of differentiation.
d
If
dx
{ φ ( x )} = f ( x ), then ∫ f ( x ) dx = φ ( x ) + C , where C is called the
constant of integration or arbitrary constant.
Symbols f ( x ) → Integrand
f ( x )dx → Element of integration
∫ → Sign of integral
φ ( x ) → Anti-derivative or primitive or integral of function f ( x )
The process of finding functions whose derivative is given, is called
anti-differentiation or integration.
Note The derivative of function is unique but integral of a function is not unique.
Some Standard Integral Formulae
Derivatives Indefinite Integrals
n+1
d x xn + 1
= x ,n ≠ −1 ∫x dx = + C, n ≠ − 1
n
(i) n
dx n + 1 n+1
d 1 1
(ii)
dx
(log e x) =
x ∫ x dx = log e | x| + C
d x
(e ) = e x ∫e dx = e x + C
x
(iii)
dx
d ax ax
= a , a > 0, a ≠ 1 ∫ a dx = +C
x x
(iv)
dx log e a log e a
d
(v)
dx
(− cos x) = sin x ∫ sin x dx = − cos x + C
, Derivatives Indefinite Integrals
d
(vi)
dx
(sin x) = cos x ∫ cos x dx = sin x + C
d
(vii)
dx
(tan x) = sec2 x ∫ sec2 x dx = tan x + C
d
(viii)
dx
(− cot x) = cosec2 x ∫ cosec2 x dx = − cot x + C
d
(ix)
dx
(sec x) = sec x tan x ∫ sec x tan x dx = sec x + C
d
(x)
dx
(− cosec x) = cosec x cot x ∫ cosec x cot x dx = − cosec x + C
(xi)
d
(log sin x) = cot x ∫ cot x dx = log|sin x| + C
dx = − log|cosec x| + C
(xii)
d
(− log cos x) = tan x ∫ tan x dx = − log|cos x| + C
dx
= log|sec x|+ C
(xiii)
d
dx
[log (sec x + tan x)] = sec x ∫ sec x dx = log|sec x + tan x| + C
π x
= log tan + + C
4 2
(xiv)
d
dx
[log (cosec x − cot x)] ∫ cosec x dx = log = cosec x
x
|cosec x − cot x| + C = log tan +C
2
sin− 1 = dx = sin−1 + C
d x 1 1 x
(xv) dx a a −x
2 2 ∫ a −x
2 2 a
−1 −1
cos −1 = ∫ dx = cos −1 + C
d x x
(xvi) dx a a −x
2 2 ∫ a −x
2 2 a
d 1 tan−1 x 1 1 1
tan−1 + C
x
(xvii) = 2 ∫ a2 dx =
dx a a a + x2 + x2 a a
1 cot −1 x −1 −1
cot −1 + C
d 1 x
(xviii) = ∫ a2 dx =
dx a a a2 + x2 + x2 a a
d 1 −1 x 1
sec −1 + C
1 1 x
sec = ∫x dx =
(xix) dx a a x x2 − a2 x −a
2 2 a a
d 1 cosec −1 x −1 −1
cosec −1 + C
1 x
= ∫x dx =
(xx) dx a a x x2 − a2 a
x −a
2 2 a
