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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    Your browser does not support the IFRAME tag.

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  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 