INTEGRATION BY SUBSTITUTION
When integrand is a function i.e., ∫fx xdx[()] () :φφ′

Here, we put φ(x) = t, so that φ ′(x)dx = dt and in
that case the integrand is reduced to ∫ftdt(). In this
method, the integrand is broken into two factors so that
one factor can be expressed in terms of the function
whose differential coefficient is the second factor.
When integrand is the product of two factors such that
one is the derivative of the other i.e. Ifxfxdx=∫ ′() () .⋅⋅
In this case we put f (x) = t and convert it into a
standard integral.

EVALUATION OF THE VARIOUS FORMS OF

INTEGRALS BY USE OF STANDARD RESULTS

(i) lx m

ax bx c

+ dx l a

++

∫ 2 ,,where ≠≠^00

Ta k e

lx m

a

ax b m lb

a

+= ++⎛ −

⎝⎜

⎞

⎠⎟

1

2

2

2

()

(ii)

lx m

ax bx c

+ dx l a

++

∫ 2 ,,where ≠≠^00

Put z^2 = ax^2 + bx + c,

so that 22 2
2

z dz

dx

ax b dz ax b

z

⋅ =+or =()+ dx

and follow the same process as explained in (i)

above.

(iii) dx

lx m ax b

ax b z

()

,

++

∫ put +=^2

(iv) dx

lx m ax bx c

lx m

() z

,

+++

∫ 2 put +=^1

(v)

dx

lx m ax b

ax b xz x z

()

22 ,^21

++

∫ put +=οr =

(vi)

dx

xa xb

mn mn

()()

,

−−

∫ where +=^2

put x – a = z(x – b)

(vii)

dx z

()

,

linear quadratic linear

(^1) =
∫ or put as
some other variable.
(viii) dx z
quadratic linear
∫ ,,put linear=^2
or some other variable, and then follow the similar
process of integration.
(ix)
dx
ax bx c lx mx n
22 putz lxax mx nbx c
++ + +
∫ ,,^2 = ++++
2
2
INTEGRATION BY PARTS
()III⋅ =I II − ⎡ ()I II
⎣⎢
⎤
⎦⎥
∫∫dx ∫ dx ∫ d +
dx
dx dx c
Choice of Ist function and IInd function depends on
order of letters in the word ILATE
I → Inverse function
L → Logarithmic function
A → Algebraic function
T → Trigonometric function
E → Exponential function
Note : A special integral
efx fxdx fxe c
xx[]()+ ′() = ()⋅ +
∫
PARTIAL FRACTIONS AND THEIR USES IN
INTEGRATION
If the integrand is a rational function, i.e., of the
form px
qx
()
()
, where p(x) and q(x) are both polynomial
functions, depending on the nature of p(x) and q(x)
integration can be done by the following processes:
(i) I f d e g r e e (p(x)) < degree (q(x)) i.e.,
fx mx n
xaxb
()
()()
= + ,
−−
a ≠ b then we write
mx n
xaxb
A
xa
B
xb

• 
  

  −−

  
  

  −

  
  


• 
  

  ()() −
  ,^ A and B being
  constants.
  (ii) I f d e g r e e (p(x)) = degree (q(x)) or
  degree (p(x)) > degree (q(x)) of non-repeated
  linear factors, i.e., fx mx nx l
  xaxb
  ()
  ()()
  = ++,
  −−
  2
  a ≠ b
  then we write mx nx l
  xaxb
  A
  xa
  B
  xb
  2
  ++ 1
  −−
  =+
  −

  
  


• 
  

  ()() −
  (iii) If denominator q(x) contains linear factors,
  some of which are repeated, i.e., integrand is
  of the form
  px
  xaxb
  ()
  ()()
  ,
  −−^2 then we write the
  integrand as A
  xa
  B
  xb
  C
  − xb

  
  


• 
  

  −

  
  


• 
  

  ()−^2
  Note : To evaluate integral of the type
  (i)
  xA
  xkxA
  dx
  2
  422

  
  


• 
  

  ∫ ++
  Divide numerator and denominator by
  x^2 and substitute x A
  x
  − =u, A being any positive
  constant.