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 ![](img64th.gif)
![](image345.gif)
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 :
![](img64th2.gif)
This technique is often compared with the chain - rule of derivatives since they both apply to composite functions.
Example 29
Evaluate ![](imgex29.gif)
Solution : ![](imgex29sol1.gif)
![](imgex29sol2.gif)
Here the inner function of the composite function ![](imgex29sol3.gif)
![](imgex29sol4.gif)
![](imgex29sol5.gif)
Example 30
Evaluate ![](imgex30.gif)
Solution : Let 4 - 3x = t,
then -3 = dt
![](imgex30sol1.gif)
Example 31
Evaluate ![](imgex31.gif)
Solution : Let ![](imgex31sol1.gif)
![](imgex31sol2.gif)
Example 32
Evaluate ![](imgex32.gif)
Solution : Let log x = t
![](imgex32sol1.gif)
Example 33
Evaluate ![](imgex33.gif)
Solution : Let ![](imgex33sol1.gif)
then -2 x dx = dt
\ 2x dx = -dt
![](imgex33sol2.gif)
Example 34
Evaluate ![](imgex34.gif)
Solution : Let arc tan x = t
![](imgex34sol1.gif)
Example 35
Evaluate ![](imgex35.gif)
Solution : Let arc sinx = t
![](imgex35sol1.gif)
Example 36
Evaluate ![](imgex36.gif) Solution : Let log x = t
Then ![](imgex36sol1.gif)
![](imgex36sol2.gif)
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