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The value of underset(x rarr0)(lim)(2int...

The value of `underset(x rarr0)(lim)(2int_(0)^(cos x ) cos^(-1) (t))/(2x- sin 2 x)dx` is

A

0

B

`1//2`

C

`-1//2`

D

`2//3`

Text Solution

Verified by Experts

The correct Answer is:
C
We have,
`lim_(x to 0) (2 underset(1)overset(cosx)int cos^(-1) t dt)/( 2x -sin 2x)`
`=lim_(x to 0) (-2sin x cos^(-1) (cosx))/( 2-2cos 2x) [ "Applying L' Hospital's rule"]`
`=- lim_(x to 0) (x sin x)/(2 sin^(2)x )=-(1)/(2) limn_(x to 0) (x)/(sin x)=-(1)/(2)`
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OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. Letf :R rarr R be a continous function given by f(x+y)=f(x)f(y) "for a...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  20. The integral underset(0)overset(pi)int x f(sinx )dx is equal to

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