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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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Index

6.1 Anti-derivatives (indefinite Integral)
6.2 Integration Of Some Trignometric Functions
6.3 Methods Of Integration
6.4 Substitution And Change Of Variables
6.5 Some Standard Substitutions
6.6 Integration By Parts
6.7 Integration By Partial Fractions

Chapter 7





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