Home
Class 12
MATHS
Let f(x) be a differentiable function su...

Let `f(x)` be a differentiable function such that `f(x)=x^2 +int_0^x e^-t f(x-t) dt` then `int_0^1 f(x) dx=`

A

`1/4`

B

`-1/12`

C

`5/12`

D

`12/7`

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)=x^(2)+int_(0)^(x)e^(-t)(x-t)dt`………….1
`x^(2)+int_(0)^(x)e^(-x-t)f(x-(x-t))dt`
[Using `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx`]
`=x^(2)+e^(-x)int_(0)^(x)e^(t)f(t)dt`……………..2
Differentiating w.r.t.`x` we get
`f'(x)=2x-e^(-x)int_(0)^(x)e^(t)f(t)dt+e^(-x)e^(x)f(x)`
`=2x-e^(-x)int_(0)^(x)e^(t)f(t)dt+f(x)`
`=2x+x^(2)` [using equation 2]
`:. f(x)=(x^(3))/3+x^(2)+c`
Also `f(0)=0` [from equation 1]
or `f(x)=(x^(3))/3+x^(2)`
or `f'(x)=x^(2)+2x`
Thus `f'(x)=` has real roots. Hence `f(x)` is non monotonic.
Hence `f(x)` is many one but range is `R` and hence, is surjective
`int_(0)^(1)f(x)dx=int_(0)^(1)((x^(3))/3+x^(2))dx`
`=[(x^(4))/12+(x^(3))/3]_(0)^(1)`
`=1/12+1/3`
`=5/12`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise MATRIX MATCH_TYPE|6 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise NUMERICAL VALUE_TYPE|28 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise MCQ_TYPE|27 Videos

  • CURVE TRACING

    CENGAGE ENGLISH|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . y=f(x) is

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. int_(0)^(1)f(x)dx=

Let f:R to R be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

Let f(x) be a continuous and differentiable function such that f(x)=int_0^xsin(t^2-t+x)dt Then prove that f^('')(x)+f(x)=cosx^2+2xsinx^2

Let f:RtoR be a differntiable function satisfying f(x)=x^(2)+3int_(0)^(x)e^(-t^(3)).f(x-t^(3))dt . Then find f(x) .

If f(x)=int_0^x tf(t)dt+2, then

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 be a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3) . Find f(x) .

Let f(x) be a derivable function satisfying f(x)=int_0^x e^t sin(x-t) dt and g(x)=f '' (x)-f(x) Then the possible integers in the range of g(x) is_______

Let f :R ^(+) to R be a differentiable function with f (1)=3 and satisfying : int _(1) ^(xy) f(t) dt =y int_(1) ^(x) f (t) dt +x int_(1) ^(y) f (t) dt AA x, y in R^(+), then f (e) =

Home
Profile