Home
Class 12
MATHS
If f(x) is an integrable function on [(p...

If f(x) is an integrable function on `[(pi)/(6),(pi)/(3)]` and
`I_(1)=int_(pi//6)^(pi//3) sec^(2)thetaf(2 sin 2theta)d theta" and "I_(2)=int_(pi//6)^(pi//3) "cosec"^(2)theta f(2 sin2theta)d theta`, then

A

`I_(1)=2I_(2)`

B

`I_(1)=3I_(2)`

C

`2I_(1)=I_(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
Using `underset(a)overset(b)int f(x) dx=underset(a)overset(b)int f(a+b)dx`, we obtain
`I_(1)=underset(pi//6)overset(pi//3)int sec^(2)((pi)/(2)-theta)f(2sin(pi-2pitheta)d theta`
Using `underset(a)overset(b)int f(x) dx=underset(a)overset(b)int f(a+b-x)dx`, we obtain
`I_(1)=underset(pi//6)overset(pi//3)int sec^(2)((pi)/(2)-theta)f(2sin(pi-2theta))d theta`
`rArr I_(1)=underset(pi//6)overset(pi//3)int "cosec"^(2)thetaf(2 sin2theta)d theta=I_(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

Let the be an integrable function defined on [0,a] if I_(1)=int_(pi//2)^(pi//2) cos theta f(sin theta cos ^(2) theta)d theta and I_(2)=int_(0)^(pi//2)sin 2 theta f(sin theta cos ^(2) theta)d theta then

Let f(x) be an integrable function defined on [a,b], b gt a gt 0 . If I_(1)=int_(pi//6)^(pi//3) f(tan theta+cos theta)sec^(2) theta d theta and, I_(2)=int_(pi//6)^(pi//3) f(tan theta +cot theta)cosec^(2) theta d theta , then (I_(1))/(I_(2))=

int_0 2pi |sin3theta| d theta

int_(0)^(pi//2) (sin^(2)theta)/ (1+cos theta)^(2) d theta =

int_ (0) ^ (pi / 2) ((theta) / (sin theta)) ^ (2) d theta =

The value of int_(pi//4)^(pi//2) cot theta cosec^(2) theta d theta is

OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. If k=int(0)^(1) (e^(t))/(1+t)dt, then int(0)^(1) e^(t)log(e )(1+t)dt i...

    Text Solution

    |

  2. If k in N and I(k)=int(-2kp)^(2kpi) |sin x|[sin x]dx, where [.] denote...

    Text Solution

    |

  3. The value of int(-1)^(1) ((logx+sqrt(1+x^(2))))/(x+log(x+sqrt(1+x^(2...

    Text Solution

    |

  4. If int(0)^(1) alphae^(betax^(2))sin(x+k)dx=0 for some alpha,beta in R,...

    Text Solution

    |

  5. int(0)^([x]//3) (8^(x))/(2^([3x]))dx where [.] denotes the greatest in...

    Text Solution

    |

  6. Let f(x)=In[cos|x|+(1)/(2)] where [.] denotes the greatest integer fun...

    Text Solution

    |

  7. lim(x to 0)(int(0^(x) x e^(t^(2))dt)/(1+x-e^(x)) is equal to

    Text Solution

    |

  8. If int(2x^(2))^(x^(3)) (In x)f(t) dt=x^(2)-2x+5, then f(8)=

    Text Solution

    |

  9. lim(x to 0)(int(-x)^(x) f(t)dt)/(int(0)^(2x) f(t+4)dt) is equal to

    Text Solution

    |

  10. If f(x+f(y))=f(x)+y for all x,y in R and f(0)=1, then int(0)^(10) f(10...

    Text Solution

    |

  11. If alpha,beta(beta gt alpha) are the roots of f(x)=-ax^(2)+bx+c=0 and ...

    Text Solution

    |

  12. The value of the constant a gt 0 such that int(0)^(a) [tan^(-1)sqrt(x)...

    Text Solution

    |

  13. If f(x) is a continuous function in [0,pi] such that f(0)=f(x)=0, the...

    Text Solution

    |

  14. Let f:R to R be continuous function such that f(x)=f(2x) for all x in ...

    Text Solution

    |

  15. The value of int(0)^(pi//4) (tan^(n)x+tan^(n-2)x)d(x-([x])/(1!)+([x]...

    Text Solution

    |

  16. The value of the definite integral int(t+2pi)^(t+5pi//2) {sin^(-1)(c...

    Text Solution

    |

  17. If f(x) is an integrable function on [(pi)/(6),(pi)/(3)] and I(1)=in...

    Text Solution

    |

  18. Let f(x)=lim(n to oo ) {(n^(n)(x+n)(x+(n)/(2))....(x+(n)/(2)))/(n!(...

    Text Solution

    |

  19. The total number of distinct x in [0,1] for which int(0)^(x) (t^(2))...

    Text Solution

    |

  20. For x in R,x ne 0, if y(x) is differentiable function such that xint(1...

    Text Solution

    |