Home
Class 12
MATHS
If int(0)^(x)f(t)dt=e^(x)-ae^(2x)int(0)^...

If `int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt`, then

A

`a=(1)/(3-2e)`

B

`f(x)=e^(x)-2e^(2x)`

C

`a=(1)/(e)`

D

`f(x)=e^(x)-e^(-x)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
Put `x=0rArr e^(0)-a int_(0)^(1)f(t)e^(-t)dt=0`
`rArr" "int_(0)^(1)f(t)e^(-1)dt=(1)/(a)" (i)"`
`rArr" "int_(0)^(x)f(t)dt=e^(x)-ae^(2x)(1)/(a)=e^(x)-e^(2x)`
Differentiating we get
`rArr" "f(x)=e^(x)-2e^(2x)`
From (i), we get `a=(1)/(3-2e)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If int_(0)^(x) f(t)dt=x^2+int_(x)^(1) t^2f(t)dt , then f'(1/2) is

If int_(0)^(x) f ( t) dt = x + int_(x)^(1) tf (t) dt , then the value of f(1) is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The value of x for which f(x) is increasing is

Let f be a continuous function satisfying the equation int_(0)^(x)f(t)dt+int_(0)^(x)tf(x-t)dt=e^(-x)-1 , then find the value of e^(9)f(9) is equal to…………………..

Let f(x) be a differentiable non-decreasing function such that int_(0)^(x)(f(t))^(3)dt=(1)/(x^(2))(int_(0)^(x)f(x)dt)^(3)AAx inR-{0} andf(1)="1. If "int_(0)^(x)f(t)dt=g(x)" then "(xg'(x))/(g(x)) is

Let f(x)=int_(2)^(x)f(t^(2)-3t+4)dt . Then

If x int_0^xsin(f(t))dt=(x+2)int_0^x tsin(f(t))dt ,w h e r ex >0, then show that f^(prime)(x)cotf(x)+3/(1+x)=0.

If f(x)=int_(2)^(x)(dt)/(1+t^(4)) , then