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

If `int_(0)^(x) f ( t) dt = x + int_(x)^(1) tf (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
`int_(0)^(x)f(t)dt=x+int_(x)^(1)tf(t)dt`
or `d/(dx) (int_(0)^(x)f(t)dt)=d/(dx) (x+int_(x)^(1)tf(t)dt)` [Using Leibniz's rule]
or `f(x)=10-xf(x)`
`=1-xf(x)`
`=f(1)-1/(1+x)`
`:f(1)=1/1`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Multiple)|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Comprehension)|31 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.11|6 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

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

Let f be a continuous function satisfying the equation int_(0)^(x)f(t)dt+int_(0)^(x)tf(x-t)dt=e^(-x)-1 , then find the value of e^(9)f(9) is equal to…………………..

If int_0^x(f(t))dt=x+int_x^1(t^2.f(t))dt+pi/4-1 , then the value of the integral int_-1^1(f(x))dx is equal to

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The value of x for which f(x) is increasing is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is

Let f(x) be a continuous and periodic function such that f(x)=f(x+T) for all xepsilonR,Tgt0 .If int_(-2T)^(a+5T)f(x)dx=19(ag0) and int_(0)^(T)f(x)dx=2 , then find the value of int_(0)^(a)f(x)dx .

CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
  1. (lim)(nvecoo)1/x[inty^a e^sin^(2t)dt-int(x+y)^a e^sin^(2t)dt-]i se q u...

    Text Solution

    |

  2. Let f(x) = sqrt(log(10)x^(2)).Find the set of all values of x for whi...

    Text Solution

    |

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

    Text Solution

    |

  4. 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 ...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  7. The value of int(1/e)^(tanx)(tdt)/(1+t^2)+int(1/e)^(cotx)(dt)/(t(1+t^2...

    Text Solution

    |

  8. underset(xtooo)lim(x^(2)"tan"(1)/(x))/(sqrt(8x^(2)+7x+1)) is equal to

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  11. Ifint0^1e^x^2(x-alpha)dx=0,t h e n a<alpha<2 (b) alpha<0 0<alpha<1 ...

    Text Solution

    |

  12. The function x^x decreases in the interval (0,e) (b) (0,1) (0,1/e) ...

    Text Solution

    |

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

    Text Solution

    |

  14. ("lim")(xvecoo)(x(log"x")^3)/(1+x+x^2)e q u a l 0 (b) -1 (c) 1 ...

    Text Solution

    |

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

    Text Solution

    |

  16. IfA=int0^pi(cosx)/((x+2)^2)dx ,t h e nint0^(pi/2)(sin2x)/(x+1)dxi se q...

    Text Solution

    |

  17. int0^4((y^2-4y+5)sin(y-2)dy)/([2y^2-9y+11])i se q u a lto 0 (b) 2 ...

    Text Solution

    |

  18. IF tan ( pi cos theta) = cot ( pi sin theta ) then cos ( theta ...

    Text Solution

    |

  19. L e tI1=thetaint0^1(e^xdx)/(1+x)a n dI2thetaint0^1(x^2dx)/(e^x^3(2-x^3...

    Text Solution

    |

  20. 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...

    Text Solution

    |