MAT136H1 Lecture Notes - Trigonometric Functions, Pythagorean Theorem
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7.3 Integration Techniques
Question #3 (Medium): Evaluating the Integral Using Secant Substitution
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 .
Evaluate the integral.
Since the denominator is in the form of , secant substitution is used. Let
. Based on the identity :
Use integration by parts to solve. Let and , and , then
. For the third integral, multiply by
, 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
and . Using Pythagorean theorem, . So
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