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Explain why it makes sense to try the trigonometric substitution x=secuif an integrand involves the expression x21

Short Answer

Expert verified

Ans: The domain of a2x2is precisely [-a,a], so the integral involving

a2x2are well suited for trigonometric substitution with x=asinu.

Step by step solution

01

Step 1. Given information.

given expression,

x2-1

02

Step 2.  Solution:-

If x=asinu, then xmust be between -aand a. Therefore the substitution x=asinumakes sense only for integrands whose domain is contained in[-a,a].

The domain of a2x2is precisely [-a,a], so the integral involving a2x2are well suited for trigonometric substitution with x=asinu.

Using the Pythagorean identity a2sin2u+a2cos2u=a2

a2x2=a2a2sin2u=a2cos2u=|acosu|=acosu

In the second last step absolute value appear because for any number A,A2=|A|.

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