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Some Standard Formulae Using Substitution

  1. 92698.png = 92692.png
  2. 92685.png = 92678.png
  3. 92665.png
  4. 92659.png
  5. 92653.png= 92647.png
  6. 92641.png = log(x + 92634.pngc
  7. 92628.png = 92622.png
  8. 92616.png
     
    = 92610.png
  9. 92604.png
     
    = 92598.png
Form Working rule
92584.png
92578.png
Divide numerator and denominator both by cos2 x, replace sec2 x, if any, in denominator by (1 + tan2 x) and put tan x = t. So that sec2 x dx = dt
92572.png
where mn ∈ N
  • If one of them is odd, then substitute for the term of even power
  • If both are odd, substitute either of them
  • If both are even, use trigonometric identities only
  • If m and n are rational numbers and 93547.png is a negative integer, thensubstitute cot x = p or tan x = p
92559.png Write sin x and cos xin terms of tan(x/2) and then substitute for tan (x/2) = t
92553.png
In this integral express numerator asλ (denominator) + μ(differentiation of denominator) + γ.
 
Find λμ, and γ by comparing coefficients of sin x, cos x and constant term and split the integral into sum of three integrals.
92529.png
92523.png
This linear factor (pxq) is expressed in terms of the derivative of the quadratic factor ax2 +bx + c together with a constant as px + q =λ(d/dx) {ax2 + bx +c} + μ ⇒ px + q =λ(2ax + b) + μ.
 
Here we find λ and μby comparing coefficients
92511.png Substitute 92505.png
92499.png Substitute 92493.png




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