∫
[
f
(
x
)
]
n
f
'
(
x
)
xⅆx
*(where n is +ve / –ve)
Put
f
(
x
)
=
t
⇒
dt=f
'
(
x
)
xⅆx
----------------------------------------------------------------------
-
∫
(
ax +b
) (
cx +d
)
ndx
*(where n is +ve /–ve)
Write
ax +b=λ
(
cx +d
)
+μ
; Put
(
cx +d
)
=
t
----------------------------------------------------------------------
-
∫
xⅆx
a x
2
+bx +c
,∫
xⅆx
√
a x
2
+bx +c
,∫
√
a x
2
+bx+c dx
Express
(a x
2
+bx +c)
=
(z2± α 2)
; α =
constant
----------------------------------------------------------------------
-
∫
(px+q)
a x2+bx +cdx
, ∫
(px +q)
√
a x2+bx +c
dx
,
∫
(px+q)
√
a x
2
+bx +c dx
Write
(px+q)= A
{
ⅆx
xⅆx
(
a x
2
+bx +c
)
}
+B
----------------------------------------------------------------------
-
∫
sinpxcosqx dx
If atleast one is odd of
p
&
q
⇒ spare one of which is odd, and substitute
the other as z and write whole in terms of
z
If both are even of
p
&
q
⇒ Use sin 2A, cos 2A formulas to simplify
If (
p+q) is –ve even integer
⇒ divide numerator & denominator by sin
kx (or cos kx); k =
–
(m+n) , then
substitute cot x = z (or tan x = z)
----------------------------------------------------------------------
-
∫
sin mx sin nx dx
, ∫
sin mx cos nx dx
,
or ∫
cos mx cos nx dx
Use : 2
sin Asin B
=
cos (A−B)−cos (A+B)
2
sin Acos B
=
sin
(
A+B
)
+sin
(
A−B
)
2
cos Asin B
=
sin
(
A+B
)
−sin
(
A−B
)
2
cos Acos B
=
cos (A−B)+cos (A+B)
----------------------------------------------------------------------
-
∫
xⅆx
asin
2
x+bcos
2
x+c
, ∫
xⅆx
a+bsin
2
x
,
∫
xⅆx
a+bcos2x
, ∫
xⅆx
(
asin x+bcos x
)
2
*(where c may be zero)
Divide numerator & denominator by
cos
2
x
Put
tan x=z
∫
xⅆx
asin x +bcos x +c
, ∫
xⅆx
a+bsin x
, ∫
xⅆx
a+bcos x
*(where c may be zero)
Put
sin x=
2 tan x
2
1+tan
2
x
2
and
cos x=
1−tan
2
x
2
1+tan
2
x
2
Simplify, then put
tan
x
2
¿z
⇒
dz=1
2sec
2
x
2xⅆx
----------------------------------------------------------------------
-
∫
asin x +bcos x+c
lsin x+mcos x+nxⅆx
For
c = n = 0
,
⇒ Write numerator =
A
{
ⅆx
xⅆx
(deno)}
+B
(deno)
For non-zero
c
&
n
,
⇒ Write numerator =
A
{
ⅆx
xⅆx
(deno)}
+B
(deno) + C
----------------------------------------------------------------------
-
Method of Partial Integration:
∫
P(x)
Q(x)
xⅆx
where degree of
P
(
x
)
<Q(x)
Break
Q(x)
into factors
(
x−a
)
,
(
x−b
)
,
(
x−c
)
…
Write
P(x)
Q(x)
=
A
(x−a)
+
B
(x−b)+C
(x−c)
*deno =
denominator
