Home
Class 12
MATHS
If a continuous function f on [0, a] sat...

If a continuous function `f` on `[0, a]` satisfies `f(x)f(a-x)=1, a >0,` then find the value of `int_0^a(dx)/(1+f(x))`

Text Solution

Verified by Experts

The correct Answer is:
`a//2`
`f(x)f(a-x)=1` or `f(a-x)=1/(f(x))`
Now, `I=int_(0)^(a)(dx)/(1+f(x))=int_(0)^(a)(dx)/(1+f(a-x))`
`=int_(0)^(a)(dx)/(1+1/(f(x)))`
`=int_(0)^(a)(f(x)dx)/(1+f(x))`
`:. 2I=int_(0)^(a)(1+f(x))/(1+f(x))dx=a` or `I=a//2`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SCQ_TYPE|113 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLE_TYPE|20 Videos

  • CURVE TRACING

    CENGAGE PUBLICATION|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

A continuous real function f satisfies f(2x)=3f(x)AAx in RdotIfint_0^1f(x)dx=1, then find the value of int_1^2f(x)dx

If f(0)=1,f(2)=3,f^'(2)=5 ,then find the value of int_0^1xf^('')(2x)dx

If fa n dg are continuous function on [0,a] satisfying f(x)=f(a-x)a n dg(x)+g(a-x)=2, then show that int_0^af(x)g(x)dx=int_0^af(x)dxdot

If f(x)=|x-1| , then the value of int_(0)^(2)f(x) dx is -

Let f be a non-constant continuous function for all xge0 . Let f satisfy the relation f(x)f(a-x)=1 for some ainR^+ . Then I=int_0^a(dx)/(1+f(x)) isd equal to

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

Let a function f(x) satisfies f(x)+f(2x)+f(2-x)+f(1+x)=x ,AAx in Rdot Then find the value of f(0)dot

Let f:RtoR 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: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)

let T le0 be a fixed real number suppose f is continuous function such that for all x in R f(x+T)=F(x) if I=int_0^T f(x)dx then the value of int_3^(3+3T)f(2x)dx is