IIT JEE , AIEEE Forum - Mathematics , Integral Calculus - Some very exclusive and extremely tough Integration problems

nitish0991

Q.1 Let f(x) x

0 be a non -ve continuous function. If f `(x).cosx

f(x).sinx

x

0
,then the value of f(5

/3) = .....
a)e^{-1/ 3} b)

3+1 c)

3-1 d)0
2 2

Q.2 The natural no. n

5 for which I_n=e^x(x-1)ⁿ.dx = 16-6e is.......
a) 1 b)2 c)3 d)4

Please help me how to solve these two question...
Thnks..

ramyadiamond

Hey for the second question, there might be limits included for the integral. Or are u sure this is the complete question??

abhujabal

Yes probably u want to ask _[0]^[1] e^x(x-1)ⁿdx

When u integrate it by parts,u would get

[e^x(x-1)ⁿ- e^xn(x-1)^n-1 - e^xn(n-1)(x-1)^n-2 ......]₀¹

So a term will come -n! e + k where k is constant

Given integral was 16 - 6e

So n! = 6

So n=3 for sure

Please rate ......

shivramkashyap

f ' (x)cosx - f(x)sinx<=0

d(f(x)cosx)<=0

F(x)=f(x)cosx is decreasing

5pi/3=2pi-pi/3>pi/2

F(5pi/3)<=F(pi/2)

f(5pi/3)cos5pi/3<=0 (cospi/2=0)

f(5pi/3)*(1/2)<=0

f is always>=0

f(5pi/3)=0 is the reqd solution

please rate if u liked it

nitish0991

both answers are wrong...answer to the 1 is c and 2 ka i myself dunoo but the question is correct...

raymonttcherian

the second question has missing limits cuz as u urself can c .. it is independent of x which can happen only when it is a DI

astrojith

Can anyone post a detailed solution for the 1st question ?