A beautiful problem involving the concept of relation-mapping from IIT JAM 2014.

Let be a differentiable function such that for all and . Then equals

The above problem can be done in many ways we will try to solve this by the simplest method.

Now, as the function is given as

So first try to change this into . Try this. It's very easy !!!

To change into we can easily do

Now we have to find the value of so we have to change the second degree term, i.e., into some linear form. Can you cook this up ???

Let us assume

i.e.,

Now you know from previous knowledge that integration is also known as anti-derivative. So can be changed into by integrating it with respect to . Try to do this integration and we are half way done !!!

On integrating both side w.r.t we get :

, (where is a integrating constant.)

Now we find the value to

We know

i.e.,

Can you find the answer now ?

Now simply, putting

we get

A beautiful problem involving the concept of relation-mapping from IIT JAM 2014.

Let be a differentiable function such that for all and . Then equals

The above problem can be done in many ways we will try to solve this by the simplest method.

Now, as the function is given as

So first try to change this into . Try this. It's very easy !!!

To change into we can easily do

Now we have to find the value of so we have to change the second degree term, i.e., into some linear form. Can you cook this up ???

Let us assume

i.e.,

Now you know from previous knowledge that integration is also known as anti-derivative. So can be changed into by integrating it with respect to . Try to do this integration and we are half way done !!!

On integrating both side w.r.t we get :

, (where is a integrating constant.)

Now we find the value to

We know

i.e.,

Can you find the answer now ?

Now simply, putting

we get