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Let Sn=sum(k=0)^n n/(n^2+kn+k^2) and Tn=...

Let `S_n=sum_(k=0)^n n/(n^2+kn+k^2)` and `T_n=sum_(k=0)^(n-1) n/(n^2+kn+k^2)` for n=1,2,3... then(correct options may be more than one) (a)`S_n lt pi/(3sqrt(3))` b) `S_n gt pi/(3sqrt(3))`(c) `T_n lt pi/(3sqrt(3))` (d) `T_n gt pi/(3sqrt(3))`

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CENGAGE PUBLICATION-INTEGRALS-All Questions
  1. prove that |sint| is non-differentiable at t=pi.

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  2. Find the value of int0^1x(1-x)dx

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  3. Let Sn=sum(k=0)^n n/(n^2+kn+k^2) and Tn=sum(k=0)^(n-1) n/(n^2+kn+k^2) ...

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  4. The value of the definite integral int(sqrt2-1)^(sqrt(2)+1)(x^4+x^2+2...

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  5. If a continuous function f on [0, a] satisfies f(x)f(a-x)=1, a >0, the...

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  6. L e tJ=int(-5)^(-4)(3-x^2)tan(3-x^2)dx and K=int(-2)^(-1)(6-6x+x^2) t...

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  7. Let g(x) be differentiable on R and int(sint)^1x^2g(x)dx=(1-sint), wh...

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  8. Show that ("lim")(n vec oo)(1/(n+1)+1/(n+2)++1/(6n))=log6

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  9. int(-pi/3)^0[cot^(-1)(2/(2cosx-1))+cot^(-1)(cosx-1/2)]dx i s equal to...

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  10. Ifint(-pi/4)^((3pi)/4)(e^(pi/4)dx)/((e^x+e^(pi/4))(sinx+cosx))=kint(-p...

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  11. If f(x) =(e^x)/(1+e^x), I1=int(f(-a))^(f(a)) xg(x(1-x))dx, and I2=int(...

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  12. If f(y)=e^y ,g(y) = y>0,a n d \ F(t)=int0^t f(t-y)g(y)dy ,then

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  13. Given int0^(pi/2)(dx)/(1+sinx+cosx)=log2. Then the value of integral ...

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  14. "I f"I1=int(-100)^(101)(dx)/((5+2x-2x^2)(1+e^(2-4x))) "and"I2=int(-10...

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  15. int0^oo(x dx)/((1+x)(1+x^2)) is equal to (A) pi/4 (B) pi/2 (C) p...

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  16. int0^oo(dx)/([x+sqrt(x^2+1)]^3)is equal to (a)3/8 (b) 1/8 (c) -3/8 ...

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  17. The value of int1^e (1+x^2 In x)/(x+x^2 In x) dx is equal (a) In (1+...

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  18. The value of integral int0^(1/sqrt3) (dx)/((1+x^2)sqrt(1-x^2)) must be...

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  19. Ifint0^oo(x^2dx)/((x^2+a^2)(x^2+b^2)(x^2+c^2)) =pi/(2(a+b)(b+c)(c+a...

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  20. The value of the integral int0^1(dx)/(x^2+2xcosalpha+1) is equal to ...

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