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Iff(x)=1+1/xint1^xf(t)dt ,t h e nt h ev ...

`Iff(x)=1+1/xint_1^xf(t)dt ,t h e nt h ev a l u eof(e^(-1))i s` 1 (b) 0 (c) `-1` (d) none of these

A

`1`

B

`0`

C

`-1`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given `xf(x)=x+int_(1)^(x)f(t)dt`…………….1
`f(x)+xf'(x)=1+f(x)`
`=f'(x)=1/x`
or `f(x)=log|x|+c`
`f(1)=1`
or `f(x)=log|x|+1`
or `f(e^(-1))=0`
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CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
  1. Let f(x) =int1^x e^t/tdt,x in R^+ . Then complete set of valuesof x f...

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  2. If int(0)^(x)f(t)dt=x+int(x)^(1)f(t)dt,then the value of f(1) is

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  3. Iff(x)=1+1/xint1^xf(t)dt ,t h e nt h ev a l u eof(e^(-1))i s 1 (b) 0 ...

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  4. If [f((sqrt(3))/2)] is [.] denotes the greatest integer function) 4 (b...

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  5. f(x) is continuous function for all real values of x and satisfies int...

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  6. the value of int(1/e->tanx) (tdt)/(1+t^2) + int(1/e->cotx) (dt)/(t*(1+...

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  7. lim(xto oo) (int(0)^(x)tan^(-1)dt)/(sqrt(x^(2)+1)) is equal to

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  8. A function f is defined by f(x) = int0^pi cos t cos(x-t)dt,0 <= x <= 2...

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  9. If f' is a differentiable function satisfying f(x)=int(0)^(x)sqrt(1-f^...

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  10. If int 0^1e^(x^2)(x-alpha)dx=0, then

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  11. The value of the integral int0^1e^x^2dx lies in the interval (0,1) ...

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  12. Given that f satisfies |f(u)-f(v)|lt=|u-v|forua n dv in [a , b]dot The...

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  13. The value of the integral int0^oo(xlogx)/((1+x^2)^2)dxi s 0 (b) log ...

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  14. int0^oo(pi/(1+pi^2x^2)-1/(1+x^2))logxdxi se q u a lto -pi/21npi (b) ...

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  15. If A=int0^pi cosx/(x+2)^2 \ dx, then int0^(pi//2) (sin 2x)/(x+1) \ dx ...

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  16. int(0)^(4)((y^(2)-4y+5)sin(y-2)dy)/([2y^(2)-8y+1]) is equal to

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  17. int(sin theta)^(cos theta) f(x tan theta)dx (where theta!=(npi)/2,n ep...

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  18. If I1=int0^1(e^x)/(1+x) dxaand I2=int0^1 x^2/(e^(x^3)(2-x^3)) dx then ...

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  19. L e tI1=int(-2)^2(x^6+3x^5+7x^4)/(x^4+2)dxa n d I2=int(-3)^1(2(x+1)^2...

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  20. Let f beintegrable over [0,a] for any real value of a. If I(1)=int(0...

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