(ii) xA

xkxA

dx

2

422

−

∫ ++

Divide numerator and denominator by x^2

and substitute x A

x

+=t, A being positive

constant.

(iii) ax bx c

px qx r

dx

2

2

++

∫ ++

put ax^2 + bx + c = l(px^2 + qx + r) + m

d

dx()px qx r n

⎛ (^2) ++
⎝⎜
⎞
⎠⎟+^
Find l, m and n by equating coefficients of
like powers of x and then split the integral
into three integrals.
TRIGONOMETRIC INTEGRALS
To find the integral axb x
cxd x
sin cos dx
sin cos

• 
  

  ∫ +
  Put a sinx + b cosx
  = L (Denominator) +M (Derivative of denominator)
  Note : To evaluate the integration of the forms
  dx
  axb x
  dx
  ab x
  dx
  (^22) sin 2 2cos (^2) ab (^2) x
  ,
  sin
  ,
  cos
  ,
  ∫∫+++∫
  dx
  axb x
  dx
  ∫( sin +++cos ) 222 ∫absin xccosx
  and
  Step 1 : Divide by cos^2 x in each case.
  Step 2 : Put tanxt dt
  at bt c

  
  

  ∫ ++
  to get the form 2 which
  is already discussed.
  INTEGRALS OF THE FORM
  (i) dx
  ∫axb xsin + cos
  (ii)
  dx
  ∫ab x+ sin
  (iii)
  dx
  ∫ab x+ cos (iv)
  dx
  ∫axb xcsin ++cos
  For all the cases (i), (ii), (iii) and (iv), universal
  substitutiontan , sin tan( / )
  tan ( / )
  x tx x
  2 x
  22
  122

  
  


• 
  

  &
  cos tan ( / )
  tan ( / )
  x x
  x
  = −

  
  


• 
  

  12
  12
  2
  2 are used. This substitution convert
  the integrals in the form dt
  ∫at (^2) ++bt c
  .
  In (i), (ii) and (iii); if a = b, then they becomes
  11
  1
  1
  a 1
  dx
  xxa
  dx
  xa
  dx
  sin cos x
  ,
  sin
  ,
  ∫∫+++∫ cos
  FUNDAMENTAL THEOREM ON CALCULUS
  I. Let f be a continuous function on closed
  interval [α, β] and A(x) be the area of function.
  Then A′(x) = f(x) ∀ x ∈ [α, β]
  II. Let f be the continuous function on closed
  interval [α, β] and F be an anti-derivative of f.
  Then ∫fx dx Fx() =[]()αβ=F() ( )−F
  α
  β
  βα
  DEFINITE INTEGRAL AS THE LIMIT OF A SUM
  An alternative method of finding fx dx()
  α
  β
  ∫ is that
  the definite integral fx dx()
  α
  β
  ∫ is a limiting case of
  the summation of an infinite series provided f(x) is
  continuous on [α, β], i.e.,
  fxdx( ) lim [ ( )h hf f( h) ... f( (n ) )]h
  α
  β
  ∫ =+++++→ 0 αα α−^1
  where h
  n
  =βα−
  PROPERTIES OF DEFINITE INTEGRALS
  (i) ∫∫f x dx() = f t dt()
  α
  β
  α
  β
  (ii) ∫∫αfx dx() =− fx dx()
  β
  β
  α
  (iii) (a) ∫∫∫fxdx() =+fxdx() fxdx() , <<
  α
  β
  α
  γ
  γ
  β
  whereαγβ
  (b) fx dx fx dx fx dx fx dx
  c
  c c
  c
  n
  ∫∫() =+ ++() ∫ () ...∫()
  α
  β
  α
  1 β
  1
  2
  (iv) ∫∫f x dx() = f(−x dx)
  00
  αα
  α
  (v) ∫∫f x dx() =+f( −x dx)
  α
  β
  α
  β
  αβ
  (vi) fx dx
  fxdx f x fx
  fx fx
  ()
  () , ( ) ()
  ,()()

  
  

  − =
  − =−
  ⎧
  ⎨
  ⎪
  ⎩⎪
  −∫
  ∫
  α
  α
  α
  2
  0
  0
  if
  if
  (vii) fxdx fxdx f x fx
  fxfx
  ()
  () , ( ) ()
  ,()()

  
  

  − =
  − =−
  ⎧
  ⎨
  ⎪
  ⎩⎪
  ∫ ∫
  0
  2
  0
  22
  02
  α α α
  α
  if
  if
  (viii) If f(t) is an odd function then gx ftdt
  a
  x
  ()=∫ ()
  is an even function.
  (ix) If f(t) is an even function then gxftdt
  x
  ()=∫ ()
  is an odd function.^0