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Standard Substitutions


 For terms of the form x2 + a2 or √x2 + a2, put x = a tanθ or a cotθ

  For terms of the form x2 - a2 or √x2 – a2 , put x = a sec θ or a cosecθ

  (A)For terms of the form a2 - x2 or √x2 + a2, put x = a sin θ or a cosθ

• If both √a+x, √a–x,  are present, then put x = a cos θ.

• For the type √(x–a)(b–x), put x = a cos2θ + b sin2θ

• For the type (√x2+a2±x)n or (x±√x2–a2)n, put the expression within the bracket = t.

• For the type  2252_equation.JPG (n ∈ N, n> 1), put x+b/x+a = t

• For 1/(x+a)n1 (x+b)n2, n1,n2 ∈ N (and > 1), again put (x + a) = t (x + b)

Example -6: Evaluate 99_integration.JPG

Solution:       623_integration.JPG= I2

                      Put 1 –x = t2  

                      dx = 2t dt

                       I = –√1–x + √x√1–x + sec–1 √1–x + k 

Example -7: Evaluate .∫ dx / (x+1)6/5 (x–3)4/5


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