Home
Class 12
MATHS
Let f(x) and phi(x) are two continuous f...

Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfying `g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0`, and `int_(b)^(2k)g(t)dt` is independent of `b`
Least positive value fo `c` if `c,k,b` are n A.P. is

A

`0`

B

`1`

C

`alpha`

D

`2alpha`

Text Solution

Verified by Experts

The correct Answer is:
D
`g(x+alpha)+g(x)=0`
or `g(x+2alpha)+g(x+alpha)=0`
or `g(x+2alpha)=g(x)`
Thus `g(x)` is periodic with period `2alpha`.
`:. int_(b)^(2k)g(t)dt=int_(b)^(b+c)g(x)dx` ( `:'b,k,c` in A.P.)
This is independent of b. Then c has least value `2alpha`.
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MATRIX MATCH_TYPE|6 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise NUMERICAL VALUE_TYPE|28 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MCQ_TYPE|27 Videos

  • CURVE TRACING

    CENGAGE PUBLICATION|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an odd function, then

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an even function, then

Let f:RtoR be a continuous function which satisfies f(x)=int_(0)^(x)f(t)dt . Then the value of f(log_(e)5) is

If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and positive constant a such that int_b^(c+b)f(x)dx is independent of b , then find the least positive value of c

If f(x) = |x-a| phi (x) , where phi(x) is continuous function, then

Let g(x) = int_(x)^(2x) f(t) dt where x gt 0 and f be continuous function and 2* f(2x)=f(x) , then

Let f : R rarr R be a continuous function which satisfies f(x) = int_0^x f(t) dt . Then the value of f(log_e 5) is

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) .

If fa n dg are continuous function on [0,a] satisfying f(x)=f(a-x)a n dg(x)+g(a-x)=2, then show that int_0^af(x)g(x)dx=int_0^af(x)dxdot

Let f be an odd continous function which is periodic with priod 2 if g(x) = int_0^x f(t) dt then

CENGAGE PUBLICATION-DEFINITE INTEGRATION -LC_TYPE
  1. Let f(x) and phi(x) are two continuous function on R satisfying phi(x)...

    Text Solution

    |

  2. Let f(x) and phi(x) are two continuous function on R satisfying phi(x)...

    Text Solution

    |

  3. Let f(x) and phi(x) are two continuous function on R satisfying phi(x)...

    Text Solution

    |

  4. Find the area of a parallelogram whose adjacent sides are given by th...

    Text Solution

    |

  5. The value of int(0)^(1)(x^(a)-1)/(logx)dx is

    Text Solution

    |

  6. Evaluating integrals dependent on a parameter: Differentiate I with ...

    Text Solution

    |

  7. The value of (dI)/(da) when I=int(0)^(pi//2) log((1+asinx)/(1-asinx)) ...

    Text Solution

    |

  8. Evaluating integrals dependent on a parameter: Differentiate I with ...

    Text Solution

    |

  9. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt The range of f(x) is

    Text Solution

    |

  10. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt f(x) is not invertibl...

    Text Solution

    |

  11. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt The value of int(0)^(...

    Text Solution

    |

  12. Let u=int0^oo (dx)/(x^4+7x^2+1 and v=int0^oo (x^2dx)/(x^4+7x^2+1) then...

    Text Solution

    |

  13. Let u=int0^oo (dx)/(x^4+7x^2+1 and v=int0^oo (x^2dx)/(x^4+7x^2+1) then...

    Text Solution

    |

  14. If f(x)=int(0)^(1)(dt)/(1+|x-t|),x in R. The value of f'(1//2) is equa...

    Text Solution

    |

  15. If f(x)=int(0)^(1)(dt)/(1+|x-t|),x in R. The value of f'(1//2) is equa...

    Text Solution

    |

  16. Let f be a differentiable function satisfying int(0)^(f(x))f^(-1)(t)d...

    Text Solution

    |

  17. Let f be a differentiable function satisfying int(0)^(f(x))f^(-1)(t)d...

    Text Solution

    |

  18. If Un=int0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zer...

    Text Solution

    |

  19. If Un=int0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zer...

    Text Solution

    |

  20. Assertion : Millikan's experiment established that electric charge is ...

    Text Solution

    |