6.4 Substitution and Change of Variables
In this method, we transform the integral to a standard form by changing the variable x of the integration to t by means of a suitable substitution of the type x = f ( t ).
Theorem : If x = f
( t ) is a differentiable function of ‘t’,
Then
Note : Comparing with , we observe that dx is replaced
by . Hence supposing
dx and dt as if they were separate entities, we have the following
working rule :
This technique is often compared with the chain  rule of derivatives since they both apply to composite functions.
Example 29
Evaluate
Solution :
Here the inner function of the composite function
Example 30
Evaluate
Solution : Let 4  3x = t,
then 3 = dt
Example 31
Evaluate
Solution : Let
Example 32
Evaluate
Solution : Let log x = t
Example 33
Evaluate
Solution : Let
then 2 x dx = dt
\ 2x dx = dt
Example 34
Evaluate
Solution : Let arc tan x = t
Example 35
Evaluate
Solution : Let arc sinx = t
Example 36
Evaluate Solution : Let log x = t
Then
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