Home
Class 12
MATHS
If f(a+b-x)=f(x), then prove that in...

If `f(a+b-x)=f(x),` then prove that `int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dxdot`

Text Solution

Verified by Experts

The correct Answer is:
NA
`I=int_(a)^(b)xf(x)dx=int_(a)^(b)xf(a+b-x)dx`
( `:'` given `f(x)=f(a+b-x))`
`=int_(a)^(b)(a+b-x)f((a+b)-(a+b-x))dx`
`=int_(a)^(b)(a+b-x)f(x)dx=(a+b)int_(a)^(b)f(x)dx-I`
or `2I=(a+b)int_(a)^(b)int_(a)^(b)f(x) dx` or `I=(a+b)/2 int_(a)^(b)f(x)dx`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SCQ_TYPE|113 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLE_TYPE|20 Videos

  • CURVE TRACING

    CENGAGE ENGLISH|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE ENGLISH|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

If f(a+b-x)=f(x) , then prove that int_a^b xf(x)dx=((a+b)/2)int_a^bf(x)dxdot

Prove that int_-a^a xf(x^4)dx=0

Prove that int_-2^2 f(x^4)dx=2int_0^2 f(x^4)dx

If f is an integrable function such that f(2a-x)=f(x), then prove that int_0^(2a)f(x)dx=2int_0^af(x)dx

Prove that: int_0^(2a)f(x)dx=int_0^(2a)f(2a-x)dxdot

Prove that int_(a)^(b)f(x)dx=(b-a)int_(0)^(1)f((b-a)x+a)dx

If f(2a-x)=-f(x), prove that int_0^(2a)f(x)dx=0

If f(a+b-x)=f(x),\ t h e n\ int_a^b xf(x)dx is equal to a. (a+b)/2int_a^bf(b-x)dx b. (a+b)/2int_a^bf(b+x)dx c. (b-1)/2int_a^bf(x)dx d. (a+b)/2int_a^bf(x)dx

If f(x+f(y))=f(x)+yAAx ,y in R and f(0)=1, then prove that int_0^2f(2-x)dx=2int_0^1f(x)dx .

If f(a+b-x) =f(x) , then int_(a)^(b) x f(x) dx is equal to

CENGAGE ENGLISH-DEFINITE INTEGRATION -CAE_TYPE
  1. Prove that int0^x[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes ...

    Text Solution

    |

  2. Evaluate: int0^oo[2e^(-x)]dx ,w h e r e[x] represents greatest intege...

    Text Solution

    |

  3. If f(a+b-x)=f(x), then prove that inta^b xf(x)dx=(a+b)/2inta^bf(x)...

    Text Solution

    |

  4. The value of the integral int3^6 sqrtx/(sqrt(9-x)+sqrtx)dx is

    Text Solution

    |

  5. Find the value of int(0)^(1)root(3)(2x^(3)-3x^(2)-x+1)dx.

    Text Solution

    |

  6. Show that int0^pifx(sinx)dx=pi/2int0^pif(sinx)dxdot

    Text Solution

    |

  7. Find the value of int0^1x(1-x)^ndx

    Text Solution

    |

  8. If a continuous function f on [0,a] satisfies f(x)f(a-x)=1,agt0, then ...

    Text Solution

    |

  9. If fa n dg are continuous function on [0,a] satisfying f(x)=f(a-x)a n ...

    Text Solution

    |

  10. Find the value of int(0)^(pi//2)sin2xlogtanxdx.

    Text Solution

    |

  11. The value of int-pi^pi cos^2x/[1+a^x].dx,a>0 is

    Text Solution

    |

  12. Evaluate: int0^pi(xsinx)/(1+cos^2x)\ dx

    Text Solution

    |

  13. Evaluate int(0)^(pi)(x dx)/(1+cos alpha sin x),where 0lt alpha lt pi.

    Text Solution

    |

  14. Find the value of int(0)^(2pi)1/(1+tan^(4)x)dx

    Text Solution

    |

  15. int0^(2pi)sin^(100)xcos^(99)x dx

    Text Solution

    |

  16. For Un=int0^1x^n(2-x)^n dx ; Vn=int0^1x^n(1-x)^ndxn in N , which of ...

    Text Solution

    |

  17. Evaluate: int0^pi log(1+cosx)dx

    Text Solution

    |

  18. Find the value of int0^1{(sin^(-1)x)//x}dx

    Text Solution

    |

  19. Evaluate int(-oo)^(0)(te^(t))/(sqrt(1-e^(2t)))dt

    Text Solution

    |

  20. If I1=int0^pixf(sin^3x+cos^2x)dxand I2=int0^(pi/2)f(sin^3x+cos^2x)dx ...

    Text Solution

    |