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If Un=int0^pi (1-cosnx)/(1-cosx) dx wher...

If `U_n=int_0^pi (1-cosnx)/(1-cosx)` dx where n is positive integer or zero, then show that `U_(n+2)+U_n =2U_(n+1)`. Hence deduce that `int_0^pi (sin^2ntheta)/(sin^2theta) =1/2(npi)`

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PATHFINDER-DEFINITE INTEGRATION-QUESTION BANK
  1. Evaluate int0^(pi/4) (e^(secx) (sin(x+(pi/4)))/(cosx(1-sinx)) dx

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  2. If absalt1 show that int0^pi (log(1+acosx))/cosx dx= pisin^(-1)a

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  3. If Un=int0^pi (1-cosnx)/(1-cosx) dx where n is positive integer or zer...

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  4. Prove that int0^x e^(xt)e^(-t^2)dt =e^(x^2/4) int0^xe^(-t^2/4)dt

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  5. Find a function g:R rarr R continous in [0,infty] satisfying g(0) =1 a...

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  6. Evaluate underset(nrarrinfty)lim 1/n overset(4n)underset(r=1)sum r/(sq...

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  7. If In=int0^1 x^ntan^(-1)x dx then prove that (n+1)In +(n-1)I(n-2)=(pi/...

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  8. If f(x)=x+int0^1(xy^2+x^2y) dy find f(x)

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  9. The total number of distinct x in[0,1] for which int0^x t^2/(1+t^4) dt...

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  10. The value of int(-pi/2)^(pi/2) (x^2cosx)/(1+e^x) dx is equal to

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  11. underset(nrarrinfty)lim((n+1)(n+2)....3n)/(n^(2n)))^(1/n) is equal to

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  12. int0^5 ((1/x)-1) dx= ?

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  13. The value of underset(nrarrinfty)lim {(sqrt(n+1) +sqrt(n+2) +......+sq...

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  14. if [x] denotes the greatest integer less than or equal to x then integ...

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  15. (phi)t={1, for 0lelt1 0 otherwise then int(-3000)^(3000) (underset(r...

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  16. Let f:RrarrR be a function defined by f(x)={([x],x,le,2),(0,x,gt,2):} ...

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  17. If alpha=int0^1(e^(9x+3tan^-1x))((12+9x^2)/(1+x^2))dx where tan^-1x t...

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  18. Let f: R rarr R be a continuous odd function, which vanishes exactly a...

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  19. Let f(x)=7tan^8x+7tan^6x-3tan^4x-3tan^2x for all x in (-pi/2, pi/2). T...

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  20. If f(x) + 2f(1 - x) =x^2 + 2, forall x in R, then f(x) is

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