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Show by differentiating (and then using algebra) that cotsin1xand 1x2xare both antiderivatives of 1x21x2. How can these two very different-looking functions be an antiderivative of the same function?

Short Answer

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Ans: By differentiating (and then using algebra) cotsin1xand1x2x both are antiderivatives of 1x21x2

Step by step solution

01

Step 1. Given information.

given expresiion,

cotsin1xand1x2x

02

Step 2. The objective is to show that −cot⁡sin−1⁡x is antiderivatives of 1x21−x2

The differentiation is,

cotsin1x=ddxcotsin1x=cotddxsin1x=cot11x2=1x21x2

Hence the expression is proved.

03

Step 3. The objective is to show that −cot⁡sin−1⁡x is antiderivatives of 1x21−x2

The differentiation is,

1x2x=1x2x=1x2x=1x2x2+1xx2+1x=1x21x2

Hence the expression is proved.

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Most popular questions from this chapter

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

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