Summary Integration formulae

(xxix)

LIST OF SOME USEFUL RESULTS FOR DIRECT APPLICATIONS
I Table of elementary integrals

 sin x dx   cos x  sinh x dx  cosh x
 cos x dx  sin x  cosh x dx  sinh x
 sec  sech x dx  tanh x
2 2
x dx  tan x


 cosec x dx   cot x
2
 cosech x dx   coth x
2




 sec x tan x dx  sec x  sech x tanh x dx  sech x
 cosec x cot x dx  cosec x  cosech x coth x dx  cosech x
 e dx  e
x x
 a dx  (a ) / log a
x x
e


n 1
xn 1 f ( x)
 
n
x n dx  , n  1 f ( x) f ( x)dx  , n  1
n 1 n 1

1 f ( x )
 x
dx  log x  f ( x)
dx  log x


 tan x dx  log sec x = – log cos x  cot x dx  log sin x
 cosec x dx  log tan(x / 2)  log(cosec x  cot x)
 sec x dx  log tan ( / 4  x / 2)  log(sec x  tan x)
dx 1 x dx 1 xa dx 1 ax
x 2
a 2

a
tan 1 ;
a x 2
a 2

2a
log
xa
, x  a; a 2
x 2

2a
log
ax
,xa

dx 1 x 1 x
 x( x 2
a ) 2 1/ 2

a
sec 1
a or  cosec 1
a a

dx x x
 (a 2
x ) 2 1/ 2
 sin 1
a or  cos1
a

dx x
 (a 2
x ) 2 1/ 2
 sinh 1
a or log{x + (x2 + a2)1/2}

dx x
 (x 2
a ) 2 1/ 2
 cosh 1
a or log{x  ( x 2  a 2 )1/ 2 }