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If u(n)=int(0)^((pi)/(2))(sin(2n-1)x)/(s...

If `u_(n)=int_(0)^((pi)/(2))(sin(2n-1)x)/(sinx)dxandv_(n)=int_(0)^((pi)/(2))((sinnx)/(sinx))^(2)dx` and n is an integer, prove that, `u_(n+1)=u_(n)andv_(n+1)-v_(n)=u_(n+1)`.

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The correct Answer is:
`(1)/(4)log.(4)/(c )`
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CHHAYA PUBLICATION-DEFINITE INTEGRAL -LONG ANSWER TYPE QUESTIONS
  1. int(0)^((3pi)/(4))(sinxdx)/(1+cos^(2)x)=(pi)/(4)+tan^(-1)((1)/(sqrt(2)...

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  2. int(0)^((pi)/(2))sin 2x tan^(-1)(sinx)dx

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  3. int(0)^((pi)/(4))xcosxcos3xdx=(pi+2)/(16)

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  4. If f(x) is periodic function of period t show that int(a)^(a+t)f(x)dx ...

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  5. If y=int(0)^(x)(t-1)(t-2)^(2)dt, find x for which (dy)/(dx)=0.

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  6. If int(0)^(y)e^(t)dt+int(0)^(x)cost^(2)dt=0 show that, e^(y)(dy)/(dx)+...

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  7. lim(n to oo)(1^(9)+2^(9)+3^(9)+...+n^(9))/(n^(10))

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  8. lim(nrarroo) [(1)/(n+1)+(1)/(n+2)+(1)/(n+3)+...+(1)/(n+n)]

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  9. lim(nrarroo) [(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+...+(1)/...

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  10. lim(nrarroo)[(n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+(n)/(n^(2)+3^(2))+......

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  11. lim(nrarroo) [(1^(2))/(n^(3)+1^(3))+(2^(2))/(n^(3)+2^(3))+(3^(2))/(n^(...

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  12. lim(nrarroo)[(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+(n+3)/(n^(2)+3^(2...

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  13. lim(nto oo)[(sqrt(n+1)+sqrt(n+2)+...+sqrt(2)n)/(sqrt(n^(3)))]

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  14. lim(n to oo)[(1)/(sqrt(n^(2)-1^(2)))+(1)/(sqrt(n^(2)-2^(2)))+(1)/(sqrt...

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  15. lim(nrarroo) [(1)/(n)+(sqrt(n^(2)-1^(2)))/(n^(2))+(sqrt(n^(2)-2^(2)))/...

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  16. lim(nrarroo) [(1)/(sqrtn)+(1)/(sqrt(2n))+(1)/(sqrt(3n))+...+(1)/(n)]

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  17. Show that, int(0)^(1)(1)/(x)log(1+x)dx=1-(1)/(2^(2))+(1)/(3^(2))-(1)/(...

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  18. If I(m)=int(1)^(e)(logx)^(m)dx, show that, I(m)+mI(m-1)lt3.

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  19. If u(n)=int(0)^((pi)/(4))tan^(n)xdx, show that, u(n)+u(n-2)=(1)/(n-1)(...

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  20. If u(n)=int(0)^((pi)/(2))(sin(2n-1)x)/(sinx)dxandv(n)=int(0)^((pi)/(2)...

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