Standard Methods of Integration and Special Integrals

Shri Laxmi Nidhi Institute of Mathematics where math comes alive STANDARD METHODS

OF INTEGRATION AND

SPECIAL INTEGRALS

1. Integration as anti-derivatives (reverse of differentiation) 2. Integration by substitution 



 f  x 

f

'

 x

f '  x

 f  x  dx 

n 1

n 1

c

 f  x  dx  log f  x   c e

f '  x







e

f  x

a

f  x



n

dx  2 f  x   c

f  x

f '  x  dx  e

f  x

c

a f  x f  x  dx  c log e a '

3. Integration by parts (ILATE Rule)   

 e  f  x   f  x   dx  e f  x   c x

'

x

  f  log x   f  log x   dx  x f  log x   c   x f  x   f  x   dx  x f  x   c '

e

e

e

'

Take the first function to be the function, which comes first in the word ‘ILATE’ where 1 1 I stands for the inverse t-functions like sin x, cos x etc. L stands for the logarithmic function A stands for the algebric function (like polynomials) T stands for the t-functions E stands for the exponential functions

4. Integration of algebric functions A. Rational algebric functions 1 px  q P ( x) , 2 , 2  Integration of forms 2 ax  bx  c ax  bx  c ax  bx  c When the integrand is a rational function and degree of the numerator is  degree of the denominator Perform long division  Integration using partial fraction  Integration of forms  Integration of form

x2  1 x2 1 x2 1 , 4 , 4 , 4 4 2 2 2 x  kx  1 x  kx  1 x  kx  1 x  kx 2  1

1

x( x n  1)

B. Irrational algebric functions 1 px  q 2  Forms , , , ( px  q ) ax 2  bx  c ax  bx  c ax 2  bx  c ax 2  bx  c 1 1 1 1  Forms , , , 2 2 2 (ax  b) cx  d (ax  bx  c ) px  q ( px  q ) ax  bx  c (ax  b) cx 2  d 5. Integration of trigonometric functions  Forms sin n x, cos n x, tan n x, cot n x,sec n x, cos ec n x Multiple Angle Method: If the square or cube or fourth power of a sine or cosine occurs in the integrand, it is generally useful to express the square (or cube or fourth power) of sine or cosine in a series of sines or cosines of multiple angles (by using the 2A or 3A formulas of trigonometry)  Forms

sin m x cos n x , sin mx cos nx , sin mx sin nx , cos mx cos nx

“A,B” Formulae Method: If the product of a sine and cosine or of two or more sines or of two or more cosines occurs in the integrand, it is generally useful to apply the “A,B” formulae of trigonometry and express the product in a series of sines or cosines of multiple angles.  Forms

1 1 1 1 , , , 2 2 2 2 a  b cos x a  b sin x  a sinx  b cosx  a cos x  b sin x cos x  c sin 2 x

1 a sinx  b cosx 1 1 1 , ,  Form a  b cosx a  b sinx a sinx  b cosx  c p sinx  q cosx  Form a sinx  b cosx 1 1 ,  Form cos 2 x  cos 2 sin 2 x  sin 2  Form