Home
Class 12
MATHS
Let f beintegrable over [0,a] for any re...

Let `f` beintegrable over `[0,a]` for any real value of `a`.
If `I_(1)=int_(0)^(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

A

`I_(1)=I_(2)`

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`I_(2)-I_(2)=underset(0)overset(pi//2)int (cos theta - sin 2 theta) f(sin theta+cos^(2)theta) d theta`
`rArr I_(2)-I_(2)=underset(0)overset(pi//2)int f(sin theta+cos^(2) theta) d(sin theta+ cos^(2) theta)`
`rArr I_(2)-I_(2)=underset(0)overset(pi//2)int f(t)dt=0, "where" t = sin theta +cos^(2) theta`
`rArr I_(1)=I_(2)`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

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

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|143 Videos

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test 2|56 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|87 Videos

  • DERIVATIVE AS A RATE MEASURER

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|26 Videos

Similar Questions

Explore conceptually related problems

The value of f(k)=int_(0)^(pi//2) log (sin ^(2) theta +k^(2) cos^(2) theta) d theta is equal to

int_0^(pi/4) (2sinthetacostheta)/(sin^4theta+cos^4theta)d theta

If f(x)=int_0^(pi/2)ln(1+xsin^2theta)/(sin^2theta) d theta, x >= 0 then :

Evaluate int_(0)^(pi//2) cos theta tan ^(-1) ( sin theta)d theta .

The value of the integral int_(0)^(pi//4) (sin theta+cos theta)/(9+16 sin 2theta)d theta , is

Evaluate : int (2 sin 2 theta- cos theta)/(6 - cos^(2) theta- 4 sin theta) d theta

Prove that: (a) (sin 2 theta)/(1+cos 2 theta)=tan theta (b) (1+sin 2 theta+cos 2theta)/(1+sin2 theta-cos 2 theta)=cot theta

int (d theta)/((sin theta - 2 cos theta)(2 sin theta + cos theta))

cos theta + sin theta - sin 2 theta = (1)/(2), 0 lt theta lt (pi)/(2)

The value of int_(0)^(pi//2)(log(1+x sin^(2) theta))/(sin^(2)theta)d theta,xge0 is equal to

OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. If I= int(0)^(1)(x)/(8+x^(3))dx then the smallest interval is which I ...

    Text Solution

    |

  2. Letf :R rarr R be a continous function given by f(x+y)=f(x)f(y) "for a...

    Text Solution

    |

  3. Let f beintegrable over [0,a] for any real value of a. If I(1)=int(0...

    Text Solution

    |

  4. The value of underset(x rarr0)(lim)(2int(0)^(cos x ) cos^(-1) (t))/(2x...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  8. int(0)^(x) (bt cos 4t-a sin 4t)/(t^(2))dt=(a sin 4x)/(x) "foe all "x ...

    Text Solution

    |

  9. Let f:RtoR,f(x)={:(|x-[x]|,[x] "is odd"),(|x-[x+1]|,[x] "is even"):} w...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  13. Let l(1)=int(0)^(1)(e^(x))/(1+x)dx and l(2)=int(0)^(1)(x^(2))/(e^(x^(3...

    Text Solution

    |

  14. Let f(x)={(1-|x|","|x|le1),(0","|x|gt1):} and g(x)=f(x-1)"for all" x i...

    Text Solution

    |

  15. If f(x)=(x-1)/(x+1),f^(2)(x)=f(f(x)),……..,……..f^(k+1)(x)=f(f^(k)(x)),k...

    Text Solution

    |

  16. If f:R in R be such that f(x)=sqrt(sin(cosx))+"In"(-2cos^(2) x+3 cos...

    Text Solution

    |

  17. If int(e)^(x) t f(t)dt=sin x-x cos x-(x^(2))/(2) for all x in R-{0}, t...

    Text Solution

    |

  18. If f (x)= int(0)^(x) {f(t)}^(-1) dt and int(0)^(1) {f(t)}^(-1)= sqrt2

    Text Solution

    |

  19. If f(x) is differentiable and int0^(t^2) x f(x) dx=2/5t^5, then f(4/(2...

    Text Solution

    |

  20. The value of int(-2)^2|1-x^2|dx is

    Text Solution

    |