# Thread to share Good Questions from Calculus which requires deep thinking

**Viram_2019**#476

@Azimuddin_Sheikh

Bro, what was your method? Only thing I could think of trying was reduction formula but it isnt leading me to any answer.

**Viram_2019**#480

@pratyaksh_tyagi

Is this from any Allen test?

If it is, then there should be some solution with techniques within JEE syllabus right.

Gaussian function JEE syllabus mein Hain @Azimuddin_Sheikh ?

**pratyaksh_2019**#482

A friend sent me this.

No gaussian function and integral out of syllabus. Maybe there is some other method

**Azimuddin_2019**#483

I remember this problem was in kvpy and they ask some problems of the level of isi , so we can expect to use some out of syllabus theorems for this problem

**Azimuddin_2019**#484

One thing we can also notice is that : Using binomial approximation, we have (for n>1),

(1+x^2)^n > 1+nx^2

Hence,

I_n=\int_0^1 \frac{dx}{(1+x^2)^n} < \int_0^1 \frac{dx}{1+nx^2} = \frac{1}{\sqrt{n}} \int_0^1 \frac{dp}{1+p^2}=\frac{\pi }{4\sqrt{n}} where p=\sqrt{n} x

Hence, \sqrt{n} I_n < \frac{\pi }{4} < 2 Hence, L<2

**Chirag_2019**#486

Bro you didn't change the limits?

But the conclusion will be same nevertheless