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

#369

Stirling approximation
Says that
ln(n!) ~ n(ln(n)-1)+\frac{1}{2}ln(2πn)
As n goes to infinity

1 Like
#370

Use \sim for \sim

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#371
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#372

Is the answer \frac {-2}{\sqrt e} @Tushar_Rathore

3 Likes
#373

Yup, its correct but i am curious in knowing the method @Rohan_Shinde1(if u know other than Leibnitz's)

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#374

@Rohan_Shinde1
For #335
I m getting
f(x)=(1-i)× \left( { \frac {(1+i)^n + (1-i)^n}{2} } \right)
Don't know how to proceed

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#375

Differentiate both sides wrt x and don't think that a differential equation will be formed. The question seems to be very well framed that no DE forms and we get the function directly.

@arush_kumar_singh

But that would turn the sum as a complex number which it isn't.

2 Likes
#376

@Tushar_Rathore
It's simple jee ques take e^x out from integral
And diffrentiate...

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#377

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#378

From here we get that
\phi_0=1 , \phi_1=1 , \phi_2 =-1 , \phi_3 = -1 , \phi_4=+1 , \phi_5 = +1 ....
So we have to calculate
\color{red} '^nC_0+'^nC_1-'^nC_2 - '^nC_3 +'^nC_4 +'^nC_5 - '^nC_6 - '^nC_7....

Am I doing correct???
@Rohan_Shinde1

1 Like
#379

1/3 ln6?

2 Likes
#380

Yep got the same \left(\frac 13 \ln 6\right). @Yash_Srivastava2 @Tushar_Rathore

1 Like
#381

@Tushar_Rathore
\frac{ln6}{3}
Yup
@Yash_Srivastava2

#382

Yoo! Its correct! Now let's move to jee advance(i believe😅)

3 Likes
#383

Answer is \displaystyle \frac {1}{2018} B\left(\frac {1}{2018},\frac {1}{2017}+1\right)- \frac {1}{2017} B\left(\frac {1}{2017},\frac {1}{2018}+1\right)

2 Likes
#384

Yeh B kya hota hai???

#385

Its in our jee course @Rohan_Shinde1 bro , so answer should also be related to jee
Options are
(1) 2018
(2)2017
(3)0
(4)2019

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#386

Standard Beta function

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#387

@Rohan_Shinde1
Is beta function converted to gamma function..
I think for \int _0^{ \frac π2 } sin^nx × cos^mx we use beta function...
But I don't know how to use it in algebraic quantity
@Tushar_Rathore
I think there should be more options like 1 ; -1

#388

@Tushar_Rathore 0

@arush_kumar_singh

The beta function and Gamma function do not have limited use bro. In higher studies you will find that Gamma function is used in many much topics like Hyperfactorial, G-Barne's Function , K -Function , Zeta function, Eta function, Gaussian integrals and many other similar integrals. It is also used in some very important theorems and transforms like Ramanujan's Master theorem, Laplace transforms , Mellin transform, Fourier transform, etc. It also has wide applications in physics too

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