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A function f is continuous for all x (an...

A function `f` is continuous for all `x` (and not everywhere zero) such that `f^2(x)=int_0^xf(t)(cost)/(2+sint)dtdot` Then `f(x)` is (a) `1/2 1n((x+cosx)/2); x!=0` (b) `1/2 1n(3/(x+cosx)); x!=0` (c) `1/2 1n((2+sinx)/2); x!=npi,n in I` (d) `(cosx+sinx)/(2+sinx); x!=npi+(3pi)/4,n in I`

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CENGAGE PUBLICATION-INTEGRALS-All Questions
  1. Evaluate: int(dx)/(x^(2/3)(1+x^(2/3)))

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  2. lim(x->0)1/x[inty^a e^(sin^2t)dt-int(x+y)^a e^(sin^2t)dt] is equal to

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  3. A function f is continuous for all x (and not everywhere zero) such th...

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  4. If int(cosx)^1(t^2f(t)dt=1-cosx AA x in (0,pi/2), then the value of [...

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  5. If int0^ycost^2dt=int0^(x^2)(sint)/t dt ,t h ep rov et h a t(dy)/(dx)=...

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  6. If int0^xf(t) dt=x+intx^1 tf(t)dt, then the value of f(1)

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  7. Evaluate: lim(x->oo)((int0^xe^(x^2)dx)^2)/(int0^x e^(2x^2)dx)

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  8. If f(x)=1+1/x int1^x f(t) dt, then the value of f(e^-1) is

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  9. If int0^1e^ (-x^2) dx=a , then find the value of int0^1x^2e^ (-x^2) dx...

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  10. Ifint0^(f(x))t^2dt=xcospix ,t h e nf^(prime)(9)i s -1/9 (b) -1/3 (c...

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  11. IfIn=int0^pix^nsinx dx ,t h e n fin d t h e v a l u eof I5+20 I3dot

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  12. The value of int0^1lim(n->oo)sum(k =0)^n(x^(k+2)2^k)/(k!)dx is: (a) e...

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  13. If I(n)=int(0)^(pi/2) sin^(n)x dx, then show that I(n)=((n-1)/n)I(n-2)...

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  14. If A=int0^1 x^(50)(2-2x)^(50)dx ,B=int0^1 x^(50)(1-x)^(50)dx , which ...

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  15. If In=int0^1(dx)/((1+x^2)^n); n in N , then prove that 2nI(n+1)=2^(-n...

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  16. Given that f satisfies |f(u)-f(v)|lt=|u-v| for u and v in [a , b]dot T...

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  17. IfIn=int0^1x^n(tan^(-1)x)dx ,t h e np rov et h a t (n+1)In+(n-1)I(n-2...

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  18. IfI=int(-20pi)^(20pi)|sinx|[sinx]dx(w h e r e[dot] denotes the greate...

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  19. IfIm=int1^x(logx)^mdx satisfies the relation (Im)=k-l I(m-1) then A)k...

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  20. The value of the integral int0^1sqrt((1-x)/(1+x))dx i s (a)pi/2+1 (b...

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