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±√x^2–a²)ⁿ, put the expression within the bracket = t.

• For the type   (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 

Solution:       = 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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