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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

1707_integration.JPG

To read more, Buy study materials of Indefinite integral comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.


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