some scary integrals

[kangkan]

Hi..Can anyone crack them..m going crazy with them

1. integrate sin^50(x)/[ sin^50(x)+cos^50(x)] .the limits of integration are from 0 to pie/2

2. integrate e^cot-1(sinx)/[ e^cot-1(sin x) + e^cot-1(cosx)..limits are 0 to pie/2 again..in words numerator = e raised to cot inverse of sinx  denominator= e raised to cot inverse of sin x +  e raised to cot inv of cosx


thanks a lot in advance

[Deepak_Ka]

use definite integral theorem f(x)=f(a+b-x)..that will suffice for both the ques.try them and lemme know if it works this way.

[kangkan]

Hi Deepak..i am unable to make out which theorem you are suggesting me to use

[Deepak_Ka]

Ok,I will show it for first ques,the second will follow the same.

f(x)=sin^50x/(sin^50x+cos^50x)
now applying f(x)=f(a+b-x)
f(x)=cos^50x/(cos^50x+sin^50x)

add the two
2f(x)=1
so,f(x)=1/2
on solving as integral we get answer as pie/4.

same approach can be applied to 2nd ques too.

[Deepak_Ka]

Ok,I will show it for first ques,the second will follow the same.

f(x)=sin^50x/(sin^50x+cos^50x)
now applying f(x)=f(a+b-x)
f(x)=cos^50x/(cos^50x+sin^50x)

add the two
2f(x)=1
so,f(x)=1/2
on solving as integral we get answer as pie/4.

same approach can be applied to 2nd ques too.

[kangkan]

Hi Deepak..i am sorrry,i am not able to understand how you are arriving a this
f(x)=sin^50x/(sin^50x+cos^50x)
now applying f(x)=f(a+b-x)
f(x)=cos^50x/(cos^50x+sin^50x)

The theorem f(x)=f (a+b-x)..what are a and b? i have never seen this theorem can you please give the details? thank you

[The Villain]

Kangkan a stands for lower lt and b for upper limit...now apply..!

[kangkan]

got it..thanks :)

[Gayyam]

Thank you Deepak!