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Chapter 5: Q. 9 (page 464)

Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution x=atan鈦�uwhen we see x2+a2, even though we can鈥檛 use the substitution x=asin鈦�uunless the integrand involves the square root ofa2鈭�x2? (Hint: Think about domains.)

Short Answer

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Ans: The domain or the substitution of x=sinu makes sense only for x鈭�[-1,1].

But the domain of the a2鈭�x2is(鈭�鈭�,鈭�).

On the other hand, the substitution of x=tanu makes sense only for x鈭�(鈭�鈭�,鈭�). That is everywhere that a2+x2 is defined.

Step by step solution

01

Step 1. Given information.

given,

why can we use the substitutionx=atanuwhen we seex2+a2

02

Step 2.  Solution:-

The domain or the substitution of x=sinu makes sense only for x鈭�[-1,1].

But the domain of the a2鈭�x2is(鈭�鈭�,鈭�).

On the other hand, the substitution of x=tan鈦 makes sense only for x鈭�(-鈭�,鈭�). That is everywhere that a2+x2 is defined

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