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Lecture

MAT136H1 Lecture Notes - Trigonometric Functions, Pythagorean Theorem


Department
Mathematics
Course Code
MAT136H1
Professor
all

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7.3 Integration Techniques
Trigonometric Substitution
Question #3 (Medium): Evaluating the Integral Using Secant Substitution
Strategy
If it contains the expression  then let   and using the identity  ,
simplify the expression. Then after the integral is solved, work with a right angle triangle and replace
by substituting back in the original variable .
Sample Question
Evaluate the integral.

Solution
Since the denominator is in the form of , secant substitution is used. Let  
, then

. Then:




. Simplifying:




 
. Based on the identity  :
 




Use integration by parts to solve. Let    and  , and  , then
 . Thus:








. For the third integral, multiply by 
, thus:




, since it is like taking the
derivative of  which is 
.
Move the second integral to the other side. Then:



. Divide both sides by :
 


Now, back to the trig substitution in the beginning: since  
,   , and   
, so
  and   . Using Pythagorean theorem,  . So   
.
Then:

  
 

Therefore,
 
 

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