Home
Class 12
MATHS
A function f(x) satisfies f(x)=sinx+int0...

A function `f(x)` satisfies `f(x)=sinx+int_0^xf^(prime)(t)(2sint-sin^2t)dt` is

A

`f((pi)/(6))=1`

B

`g(x)=int_(0)^(x)f(t)` dt is increasing on `(0,pi)`

C

`f(0)=0`

D

f(x) is increasing on `(0,pi)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`f(x)=sin x +int_(0)^(x)f'(t)(2 sin t - sin^(2)t)dt`
`rArr" "f'(x)=cos x+f'(x)(2 sin x-sin^(2)x)`
`rArr" "(1-2 sin x+sin^(2)x)f'(x)=cosx`
`rArr" "f'(x)=(cosx)/((sin x-1)^(2))`
`rArr" "f(x)=(1)/((1-sinx))+c`
since f(0)= 0 we have `c=-1`
`rArr" "f(x)=(sinx)/(1-sinx)`
Promotional Banner

Similar Questions

Explore conceptually related problems

A Function f(x) satisfies the relation f(x)=e^x+int_0^1e^xf(t)dtdot Then (a) f(0) 0

Consider a real valued continuous function f such that f(x)=sinx + int_(-pi//2)^(pi//2) (sinx+t(f(t))dt . If M and m are maximum and minimum values of function f , then the value of M//m is____________.

Let f(x) be a differentiable function satisfying f(x)=int_(0)^(x)e^((2tx-t^(2)))cos(x-t)dt , then find the value of f''(0) .

Let f:RtoR be a differntiable function satisfying f(x)=x^(2)+3int_(0)^(x^(1/3))e^(-t^(3)).f(x-t^(3))dt . Then find f'(x) .

Let f: R to R be a continuous function which satisfies f(x)= int_0^xf(t)dtdot Then the value of f(1n5) is______

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of lim_(x to 0)(cosx)/(f(x)) is equal to where [.] denotes greatest integer function

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of int_(0)^(pi//2) f(x)dx lies in the interval

Let f : R rarr R be a continuous function which satisfies f(x) = int_0^x f(t) dt . Then the value of f(log_e 5) is

Let f be a non-negative function defined on the interval [0,1] . If int_0^xsqrt(1-(f^(prime)(t))^2)dt=int_0^xf(t)dt ,0lt=xlt=1,a n d \ f(0)=0 , then

If a continous function f satisfies int_0^f(x) t^2 dt=x^2(1+x) for all xge0 then f(2) is equal to