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int(-pi/3)^0[cot^(-1)(2/(2cosx-1))+cot^(...

`int_(-pi/3)^0[cot^(-1)(2/(2cosx-1))+cot^(-1)(cosx-1/2)]dxi se q u a lto` `(pi^2)/6` (b) `(pi^2)/3` (c) `(pi^2)/8` (d) `(3pi^2)/8`

A

`(pi^(2))/6`

B

`(pi^(2))/3`

C

`(pi^(2))/8`

D

`(3pi^(2))/8`

Text Solution

Verified by Experts

The correct Answer is:
A
For `x epsilon(-(pi)/3,0),2 cos x-1gt0`
`:.I=int_(-pi//3)^(pi)//2 dx=(pi^(2))/6`
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CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
  1. The numbers of possible continuous f(x) defined in [0,1] for which I1...

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  2. Suppose that F (x) is an antiderivative of f (x)=sinx/x,x>0 , then...

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  3. int(-pi/3)^0[cot^(-1)(2/(2cosx-1))+cot^(-1)(cosx-1/2)]dxi se q u a lto...

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  4. Evaluate the definite integrals int0pi/4(sinx+cosx)/(9+16sin2x)dx

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  5. int- 1^1(e^(-1/ x))/(x^2(1+e^(-2/ x)))dx is equal to :

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  6. If int0^oosinx/xdx=pi/2, then int0^oosin^3x/xdx is equal to

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  7. The range of the function f(x)=int(-1)^(1)(sinxdt)/(1+2tcosx+t^(2)) is

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  8. If the function f:[0,8]toR is differentiable, then for 0ltbetalt1 and ...

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  9. If f(x)=x^(5)+5x-1 then int(5)^(41)(dx)/((f^(-1)(x))^(5)+5f^(-1)(x)) e...

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  10. Let f(0)=0a n dint0^2f^(prime)(2t)e^(f(2t))dt=5.t h e nv a l u eoff(4)...

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  11. If f(x)-3cos(tan^(-1)x), then the value of the integral int(0)^(1)xf''...

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  12. The equation of the curve is y=f(x)dot The tangents at [1,f(1),[2,f(2)...

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  13. The value of int1^e((tan^(-1)x)/x+(logx)/(1+x^2))dxi s tane (b) tan^...

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  14. If f(pi)=2 and int(0)^(pi)(f(x)+f''(x))sin x dx=5, then f(0) is equal ...

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  15. If int1^2e^x^2dx=a ,t h e ninte^(e^4)sqrt(1n x)dxi se q u a lto 2e^4-...

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  16. If f(x) is continuous for all real values of x , then sum(r=1)^nf(r-1...

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  17. The value of int0^(pi/2)sin|2x-alpha|dx , where alpha in [0,pi] , is

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  18. f(x) is a continuous function for all real values of x and satisfies i...

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  19. If f(x)=int(-1)^(x)|t|dt, then for any xge0,f(x) equals

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  20. If agt0 and A=int(0)^(a)cos^(-1)xdx, and int(-a)^(a)(cos^(-1)x-sin^(...

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