Home
Class 12
MATHS
If int(0)^(x) f(t)dt=x+int(x)^(1) t f(t)...

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,
`underset(0)overset(x)intf(t)dt=x+underset(x)overset(1)int t f(t) dt`
`rArr underset(0)overset(x)intf(t)dt=x-underset(1)overset(x)int t f(t) dt`
`rArr underset(0)overset(x)intf(t)dt=x+underset(1)overset(x)int t f(t) dt=x`
Differentiating w.r.t to x, we get
`f(x)+xf(x)=1rArr f(x)=(1)/(x+1)rArr f(1)=(1)/(2)`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Section II - Assertion Reason Type|12 Videos

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Exercise|147 Videos

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Chapter Test 2|60 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA|Exercise Exercise|86 Videos

  • DERIVATIVE AS A RATE MEASURER

    OBJECTIVE RD SHARMA|Exercise Exercise|26 Videos

Similar Questions

Explore conceptually related problems

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

f(x)=int_(0)^( pi)f(t)dt=x+int_(x)^(1)tf(t)dt, then the value of f(1) is (1)/(2)

If f:R in R is continuous and differentiable function such that int_(-1)^(x) f(t)dt+f'''(3) int_(x)^(1) dt=int_(1)^(x) t^(3)dt-f'(1)int_(0)^(x) t^(2)dt+f'(2) int_(x)^(3) r dt , then the value of f'(4), is

If f(x)=int_(0)^(x)e^(-t)f(x-t)dt then the value of f(3) is

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

If f(x)=1+(1)/(x)int_(1)^(x)f(t)dt, then the value of (e^(-1)) is

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

f(x)=x^(3)+1-2x int_(0)^(x)g(t)dt and g(x)=x-int_(0)^(1)f(t)dt. Then yhe value of int_(0)^(1)f(t)dt lies in the interval

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

OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. Let f(x) be a polynomial of degree 2 satisfying f(0)=1, f(0) =-2 and f...

    Text Solution

    |

  2. The vlaue of int(-2)^(2) (sin^(2)x)/([(x)/(pi)]+(1)/(2))dx where [*]...

    Text Solution

    |

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

    Text Solution

    |

  4. If f(x)= int0^(sinx) cos^(-1)t dt +int(0)^(cosx) sin^(-1)t dt, 0 lt ...

    Text Solution

    |

  5. Let f(x) be a continuous function such that int(n)^(n+1) f(x) dx=n^(3)...

    Text Solution

    |

  6. Let f(x)=(e^(x)+1)/(e^(-x)-1) and int(0)^(1)x^(3)(e^(x)+1)/(e^(x)-1) d...

    Text Solution

    |

  7. If int(0)^(1) x e^(x^(2) ) dx=alpha int(0)^(1) e^(x^(2)) dx, hten

    Text Solution

    |

  8. If I=int(0)^(1) (1+e^(-x^2)) dx then, s

    Text Solution

    |

  9. If I= int(0)^(1)(x)/(8+x^(3))dx then the smallest interval is which I ...

    Text Solution

    |

  10. Let f: r in R be a contunuous function given be f(x+y)=f(x)+f(y)"for a...

    Text Solution

    |

  11. Let the be an integrable function defined on [0,a] if I(1)=int(pi//2)^...

    Text Solution

    |

  12. lim(x to 0) (2 int(0)^(cosx) cos^(-1) t dt)/( 2x -sin 2x) is equal to

    Text Solution

    |

  13. If I(1)= int(1)^(sin theta) (x)/(1+x^(2)) dx and I(2) int(1)^("cosec" ...

    Text Solution

    |

  14. If f(x)=int(1)^(x) (log t)/(1+t) dt"then" f(x)+f((1)/(x)) is equal to

    Text Solution

    |

  15. Let F(x) =f(x) +f((1)/(x)),"where" f(x)=int(1)^(x) (log t)/(1+t) dt Th...

    Text Solution

    |

  16. If int(0)^(x) (bt cos 4 t - a sin 4t)/( t^(2))dt=(a sin 4x)/(x) "for a...

    Text Solution

    |

  17. Let f: R in R be given by f(x)={{:(|x-[x]|,"when[x]is odd"),(|x-[x]-...

    Text Solution

    |

  18. If f(x) = sin x +cos x and g(x) = {:{((|x|)/(x),","x ne0),(2,","x=0):}...

    Text Solution

    |

  19. If x in[(4n+1)(pi)/(2),(4n+3)(pi)/(2)] and n in N, then the value of i...

    Text Solution

    |

  20. If f:R in R is continuous and differentiable function such that int(...

    Text Solution

    |