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L e t \ f(n)=1+1/2+1/3+...+1/n. Then sho...

`L e t \ f(n)=1+1/2+1/3+...+1/n`. Then show that `f(n)=int_0^(pi/2)cot(theta/2)(1-cos^ntheta)d theta`

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CENGAGE PUBLICATION-INTEGRALS-All Questions
  1. Evaluate: int0^(1)sin^(-1)((2x)/(1+x^2))dx

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  2. Evaluate: int0^(pi/4)(tan^(-1)((2cos^2theta)/(2-sin2theta)))sec^2theta...

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  3. L e t \ f(n)=1+1/2+1/3+...+1/n. Then show that f(n)=int0^(pi/2)cot(the...

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  4. Let f(x) be a continuous function AAx in R , except at x=0, such that...

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  5. If f(x)=(sinx)/xAAx in (0,pi], prove that pi/2int0^(pi/2)f(x)f(pi/2-x)...

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  6. Determine a positive integer n such that int0^(pi/2)x^nsinx dx=3/4(pi^...

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  7. f,g,h are continuous in [0,a],f(a-x)=f(x),g(a-x)=-g(x),3h(x)-4h(a-x)=5...

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  8. If f(x+f(y))=f(x)+yAAx ,y in R and f(0)=1, then prove that int0^2f(2-...

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  9. Let intx^(x+p)f(t)dt be independent of xa n dI1=int0^pf(t)dt ,I2=int(1...

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  10. Evaluate: int3^10[log[x]]dx

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  11. If the function f : [0,8] to R is differentiable, then for 0 < alpha...

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  12. The value of int0^1(x^4(1-x)^4)/(1+x^2)dx is/are (a)(22)/7-pi (b)...

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  13. Let intx^(x+p)f(t)dt be independent of xa n dI1=int0^pf(t)dt ,I2=int(1...

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  14. Evaluate:int-pi^pi ((2x)(1+sinx)/(1+cos^2x))dx

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  15. Q. if int0^100(f(x) dx = a, then sum(r=1)^100(int0^1( f(r-1+x)dx))=

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  16. Let f(x) be a derivable function satisfying f(x)=int0^x e^tsin(x-t)dt...

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  17. If int0^1 e^(x^2)(x-alpha)dx=0, then (a)alphalt2 (b)alphalt0 (c)"" 0l...

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  18. int(5/2)^5(sqrt((25-x^2)^3))/(x^4)dx is equal to

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  19. The value of int0^((3pi)/2)(|tan^(-1)(tanx)|-|sin^(-1)(sinx)|)/(|tan^(...

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  20. Let f(0)=0a n dint0^2f^(prime)(2t)e^(f(2t))dt=5.t h e nv a l u eoff(4)...

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