TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui. 0345-3093759
INTEGRATION FORMULAE 1.
u vdx udx vdx
2.
ax dx a x dx ,
3.
n x dx
4.
ax b n ax b dx an 1
5.
x dx ln x c
6.
x a dx
7.
nx a dx
1 a nx c , where n is any constant n ln a
8.
e
1 ax e c a
9.
x a ln x a c
10.
11.
a 2 x 2 a tan
12.
x
13.
x 2 a 2 2a ln x a
14.
a 2 x 2 2a ln a x
15.
ln x x 2 a 2 c x a
16.
ln x x 2 a 2 c x a
17.
a 2 x 2 dx
1 1 x x a 2 x 2 a 2 sin 1 c 2 2 a
18.
x 2 a 2 dx
1 1 x x 2 a 2 a 2 ln x x 2 a 2 c 2 2
19.
x 2 a 2 dx
1 1 x x 2 a 2 a 2 ln x x 2 a 2 c 2 2
where a is any constant.
x n 1 c n 1 n 1
c
1
ax
dx
ax c ln a
dx
dx a x 2
2
sin 1
x c a
1
x c a
dx
dx a x 2
2
1
1 x sec1 c a a
dx
1
xa
dx
1
ax
c
c
dx
2
2
dx
2
2
TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui. 0345-3093759
20.
cos x dx sin x c
21.
cos nx dx n sin nx c
22.
sin x dx cos x c
23.
sin nx dx n cos nx c
24.
tan x dx ln sec x c
25.
cot x dx ln sin x c
26.
secx dx ln secx tan x c
27.
cosecx dx ln cosecx cot x c
28.
sec
29.
cosec x dx cot x c
30.
sec x tanx dx sec x c
31.
cos ecx cotx dx cos ecx c
1
1
2
x dx tan x c 2
Integration By Trigonometric Substitution An integrand which contains one of the forms a 2 x 2 , a 2 x 2 , x 2 a 2 but no other irrational factor, may be transformed into another involving trigonometric substitution of a new variable as follows: For
use
to obtain
a2 x2
x = a sinθ
a 1 sin 2 a cos
a2 x2
x = a tanθ
a 1 tan2 a sec
x2 a2
x = a sec θ
a sec2 1 a tan
Integration By Parts Integral of product of two functions = 1st function × Integral of the 2nd − Derivative of the 1st function × Integral of 2nd function.
du
u v dx u v dx dx v dxdx