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Prove that for any positive integer k ,...

Prove that for any positive integer `k ,(sin2k x)/(sinx)=2[cosx+cos3x++cos(2k-1)x]dot` Hence, prove that `int_0^(pi/2)sin2x kcotxdx=pi/2dot`

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CENGAGE ENGLISH-INTEGRALS-All Questions
  1. If alpha=int0^1(e^(9x+3tan^(-1)x))((12+9x^2)/(1+x^2))dx where tan^(-1)...

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  2. Evaluate int(-4)^(-5)e^((x+5)^(2))dx+3int(1//3)^(2//3)e^(9(x-2/3)^(2))...

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  3. Prove that for any positive integer k ,(sin2k x)/(sinx)=2[cosx+cos3x+...

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  4. Compute the integrals: int0^oof(x^n+x^(-n))logx(dx)/x

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  5. If for a real number y ,[y] is the greatest integral function less, th...

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  6. Compute the integrals: int0^oof(x^n+x^(-n))logx(dx)/(1+x^2)

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  7. int0^x{int0^uf(t)dx}\ du is equal to (a) int0^x(x-u)f(u)du (b) int0...

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  8. Compute the integrals: int(1/e)^e1/xsin(x-1/x)dx

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  9. STATEMENT 1: int0^pixsinxcos^2xdx=pi/2int0^pisinxcos^2x dx. STATEMEN...

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  10. Evaluate: int0^(pi/2)xcosx dx

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  11. Evaluate: int0^(pi/4)(sinx+cosx)/(9+16sin2x)dx

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  12. Evaluate: int(-pi//4)^(pi//4)log(sinx+cosx)dx

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  13. If alpha,beta(beta>alpha), are the roots of g(x)=a x^2+b x+c=0 and f...

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  14. Consider the function f(x) satisfying the relation f(x+1)+f(x+7)=0AAx ...

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  15. Let f: RvecR be a differentiable function having f(2)=6,f^(prime)(2)=1...

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  16. Ifint(sinx)^1t^2f(t)dt=1-sinx ,w h e r e .x in (0,pi/2), then find th...

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  17. Prove that inta^bf(x)dx=int(a+c)^(b+c)f(x-c)dx , and when f(x) is odd ...

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  18. Evaluate: int0^(100pi)sqrt((1-cos2x))dxdot

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  19. Evaluate: int0^pi(x dx)/(1+cosalphasinx),w h e r e0ltalphaltpidot

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  20. Evaluate : int0^(pi/4)(sinxcosx)/(sin^4x+cos^4x)\ dx

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