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If int(0) ^(x) f (t) dt = x + int (x ...

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

A

`1//2`

B

`0`

C

`1`

D

`-1//2`

Text Solution

Verified by Experts

The correct Answer is:
A
We have
` int_(0) ^(x) f (t) dt = x + int _(x) ^(1) t f (t) dt `
`rArr (d)/(dx) { int _(0) ^(x) f (t) dt } = (d)/(dx) { x + int_(x) ^(1) t f(t) dt }`
`rArr f (x) = 1 + 0 - x f (x)" "` [Using Leibnitz's rule]
` rArr f (x) = 1 - xf (x)`
`rArr f (x) = (1)/( x + 1) rArr f(1) = (1)/(2)`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. int(0)^(10)|x(x-1)(x-2)|dx is equal to

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

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  3. Let f be a positive function. Let I1 = int(1-k)^(k) xf{x(1-x}dx, I(2...

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  4. If g(x)=int(0)^(x)cos^(4) dt, then g(x+pi) equals

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  5. If l(n)=int(0)^(pi//4) tan^(n)x dx, n in N "then" I(n+2)+I(n) equals

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  6. If rArr I(n)=int(0)^(pi//4) tan ^(n)x dx, then for any positive integ...

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  7. The value of int(-1)^(1) (d)/(dx) ("tan"^(1)(1)/(x))dx is

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  8. The vlaue of the integral int(-1)^(3) ("tan"^(1)(x)/(x^(2)+1)+"tan"^...

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  9. If =int(1)^(e) (logx)^(n) dx, "then"I(n)+nI(n-1) is equal to

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  10. If =int(0)^(1) x^(n)e^(-x)dx "for" n in N "then" I(n)-nI(n-1)=

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  11. The value of int(1//n)^((an-1)//n) (sqrt(x))/(sqrt(a-x+sqrtx))dx, is

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  12. The value of the integral int(0)^(pi//2)log |tan x cot x |dx is

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  13. If I(1)=int(x)^(1)(1)/(1+t^(2)) dt and I(2)=int(1)^(1//x) dt "for" x g...

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  14. For all values of , int(1//e)^(tanx) (t)/(1+t^(2))dt+int(1//e)^(tanx) ...

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  15. The absolute value of int(10)^(19) (cosx)/(1+x^(8))dx, is

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  16. If f(x) is an odd pefiodc function defined on the interval [T/2,T/2], ...

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  17. If int(pi//2)^(theta) sin x dx=sin 2 theta then the of theta satisfyin...

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  18. If f(x) is periodic function with period, T, then

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  19. If f(n)=(1)/(n){(n+1)(n+2)(n+3)...(n+n)}^(1//n) then lim(n to oo)f(n)...

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  20. The poins of extremum of phi(x)=int(1)^(x) e^(-t^(2)//2) (1-t^(2))dt a...

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