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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)dot` Hence, deduce that `int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot`

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CENGAGE PUBLICATION-INTEGRALS-All Questions
  1. Let a+b=4,w h e r ea<2,a n dl e tg(x) be a differentiable function. If...

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  2. Evaluate: int(-pi/2)^(pi/2)log{(a x^2+b x+c)/(a x^2-b x+c)(a+b)|sinx|}...

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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. Evaluate: int(-1)^1(x^3+|x|+1)/(x^2+2|x|+1)dx

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  5. Find the value of int(-1)^(3/2)|xsinpix|dx

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  6. Evaluate: int(-pi/2)^(pi/2)sin^2xcos^2x(sinx+cosx)dx

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  7. int(-1)^2[([x])/(1+x^2)]dx ,w h e r e[dot] denotes the greater integer...

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  8. Evaluate: inta^b cos x dx

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  9. Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in R....

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  10. Evaluate: ("lim")(n to oo)((1^2+2^2+3^3++n^2)(1^3+2^3+3^3++n^3))/(1^6+...

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  11. ifint(dx)/(x^2(x^n+1)^(((n-1)/n))=-[f(x)]^(1/n)+c ,t h e nf(x)i s (1+...

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  12. Evaluate the definite integrals as limit of sums int1^2x^2dx

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  13. Evaluate: inta^bsinx dx using limit of sum

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  14. int(-3)^3 x^8{x^11}dx

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  15. Evaluate: inta^b e^x dx using limit of sum

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  16. The value of int(-2)^1[x[1+cos((pix)/2)]+1]dx , where [.] denotes the...

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  17. Evaluate: lim(n->oo)(((n+1)(n+2)...(n+n))^(1/n))/n

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  18. If g(x)=int0^x(|sint|+|cost|)dt ,then g(x+(pin)/2) is equal to, where...

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  19. Evaluate: lim(n->oo)[1/(n a)+1/(n a+1)+1/(n a+2)++1/(n b)]

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  20. int0^x|sint|dt, where x in (2npi,(2n+1)pi),ninN,is equal to (A) 4n...

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