Problems related to differentiation under the integral sign are very common in IIT-JEE. We will see how to solve such problems using Leibniz's Rule given below.
Problem 1
If f(x) is differentiable and ,
find f(4/25)
(IIT-JEE 2004)
Solution
Differentiating both sides,
Hence, f(4/25)=2/5 for t=2/5
PS: Note that -2/5 is also a valid answer. It was not among the options for the MCQ.
Problem 2
Solve
This is in 0/0 form. So we can apply L'Hospital's Rule directly. Differentiating the numerator and the denominator separately, we have,
Problem 1
If f(x) is differentiable and ,
(IIT-JEE 2004)
Solution
Differentiating both sides,
PS: Note that -2/5 is also a valid answer. It was not among the options for the MCQ.
Problem 2
Solve
(IIT-JEE 2007)
Solution This is in 0/0 form. So we can apply L'Hospital's Rule directly. Differentiating the numerator and the denominator separately, we have,
Is the first question even right? Put t=sqrt(2) and you get that integral from 2 to 2 of a continuous function is 8*sqrt(2)/5 which doesn't make sense.
ReplyDeleteAlso, the function you got is odd and not even in t. So if you use -2/5 instead of 2/5, you will get -2/5 as answer (Assuming you have solved it correctly). So the problem is wrong in so many ways :)
Well, there was a typo in the problem 1 , the lower limit of the integral is 0 and NOT 2.
ReplyDeleteAlso, -2/5 is also valid answer (Didn't put it there initially because it was not an option for the MCQ)
Thanks Raziman for pointing out..
Integral from 0 to t^2 of any function has to be an even function of t, so there is something wrong with Q1 definitely :)
ReplyDeleteIn 2nd Que, it is not in the form 0/0 or ∞/∞ so LH rule can't apply
ReplyDelete