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Question

How many of the following statements are correct?
1. 1a2x2dx=1asin1(xa) + c
2. 1|x|x21dx=sec1(x) + c
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Solution

We want to find the integrals 1a2x2 and 1|x|x21.
We will find these integrals by method of inspection, trying to guess the integral, like we did for integral of sinx, cosx and many other functions. Since we are given integrals on the RHS, to solve this question, we simply differentiate the functions on the RHS and check if we get the same function in the integral.
We know that derivative of sin1(x) is 11x2.
But we want to find integral of 1a2x2
For this we will take ‘a’ common from the denominator. So we get 1|a|1(xa)2. This may be related to the derivative of sin1(xa).
To check that, let’s find the derivative of sin1(xa). We get 1a1(xa)2, which is same as the integral we want to find.
1a2x2dx=sin1(xa)First statement is not correct.2.1|x|x21 is the derivative of sec1(x)dx|x|x21=sec1(x)

So the second statement is correct We can also find these integrals by substituting x = a sin t and x = sec t

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