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Let `f: R -> R` be a differentiable function such that `f(0)=0, f((pi/2))=3` and `f'(0)=1` If `g(x) = int_x^(pi/2) [f'(t) cosec t- cot t cosec t f(t)] dt` for `x in (0, pi/2]` then `lim_(x->0) g(x)=`

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MOTION-DEFINITE INTEGRATION -EXERCISE -4 LEVEL-II
  1. T h ev a l u eofint(sqrt(1n2))^(sqrt(1n3))(xsinx^2)/(sinx^2+sin(1n6-x^...

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  2. Let f:[1,oo] be a differentiable function such that f(1)=2. If int1...

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  3. The value of the integral int(-pi//2)^(pi//2) (x^(2) + log" (pi-x)/(pi...

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  4. Let f:[1/2,1]vecR (the set of all real numbers) be a positive, non-con...

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  5. Find a for which lim(n->oo) (1^a+2^a+3^a+...+n^a)/((n+1)^(a-1)[(na+1)+...

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  6. Let f:(0,oo)vecR be given by f(x)=int(1/x)^x(e^(-(t+1/t))dt)/t , then ...

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  7. Let f:[0,2]vecR be a function which is continuous on [0,2] and is diff...

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  8. The integral overset(pi//2)underset(pi//4)int (2 cosecx)^(17)dx is equ...

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  9. Let f: RvecR be a function defined by f(x)={[x],xlt=2 0,x >2 where [...

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  10. If alpha=int0^1(e^9x+3tan^((-1)x))((12+9x^2)/(1+x^2))dxw h e r etan^(-...

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  11. Let f: RvecR be a continuous odd function, which vanishes exactly at o...

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  12. Let f (x) = 7 tan ^(8) x + 7 tan ^(6) x - 3 tan ^(4) x - 3 tan ^(2) x ...

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  13. Let f^(prime)(x)=(192 x^3)/(2+sin^4pix)fora l lx in Rw i t hf(1/2)=0....

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  14. Let F : R to R be a thrice differentiable function . Suppose that F(...

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  15. Let F : R to R be a thrice differentiable function . Suppose that F(...

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

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  17. Let f(x)=lim(n->oo)((n^n(x+n)(x+n/2)....(x+n/n))/(n!(x^2+n^2)(x^2+n^2/...

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  18. Let f: R -> R be a differentiable function such that f(0)=0, f((pi/2))...

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  19. For each positive integer n, let jy(n) =1/n ((n +1) (n +2) …(n +n) )...

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  20. The value of the integral int0^(1/2)(1+sqrt(3))/(((x+1)^2(1-x)^6)^(...

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