( )
n
n
k
f
x dx
k
f
x dx
k
f
x dx
+
+
+
∫
∫
∫
INTEGRALS         355
� Some standard integrals
(i)
1
C
1
n
n
x
x dx
n
+
=
+
+
∫
, n ≠ – 1 Particularly, 
C
dx
=x
+
∫
(ii)
cos
sin
C
x dx
x
=
+
∫
(iii)
sin
– cos
C
x dx
x
=
+
∫
(iv)
sec2
tan
C
x dx
x
=
+
∫
(v)
cosec2
– cot
C
x dx
x
=
+
∫
(vi)
sec
tan
sec
C
x
x dx
x
=
+
∫
(vii)
cosec
cot
– cosec
C
x
x dx
x
=
+
∫
(viii)
1
2
sin
C
1
dx
x
x
−
=
+
−
∫
(ix)
1
2
cos
C
1
dx
x
x
−
= −
+
−
∫
(x)
1
2
tan
C
1
dx
x
x
−
=
+
+
∫
(xi)
1
2
cot
C
1
dx
x
x
−
= −
+
∫+
(xii)
C
x
x
e dx
=e
+
∫
(xiii)
C
log
x
x
a
a dx
a
=
+
∫
(xiv)
1
2
sec
C
1
dx
x
x x
−
=
+
−
∫
(xv)
1
2
cosec
C
1
dx
x
x x
−
= −
+
−
∫
(xvi)
1
log|
|
C
dx
x
x
=
+
∫
� Integration by partial fractions
Recall that a rational function is ratio of two polynomials of the form P( )
Q( )
x
x
,
where P(x) and Q (x) are polynomials in x and Q (x) ≠ 0 If degree of the
polynomial P (x) is greater than the degree of the polynomial Q (x), then we
may divide P (x) by Q (x) so that 
1P ( )
P( )
T ( )
Q( )
Q( )
x
x
x
x
x
=
+
, where T(x) is a
polynomial in x and degree of P1(x) is less than the degree of Q(x) T(x)
being polynomial can be easily integrated