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Prove that for any positive integer K, (...

Prove that for any positive integer K, `(sin2kx)/(sinx)=2[cos x + cos 3x+....+ cos (2k-1)x]` Hence , proved that ` int_(0)^(pi//2) sin 2kx. Cot x dx = (pi//2)`

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The correct Answer is:
`pi/2`
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MOTION-DEFINITE INTEGRATION -EXERCISE -4 LEVEL-II
  1. Prove that for any positive integer K, (sin2kx)/(sinx)=2[cos x + cos 3...

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  2. Let y =f (x) be a twice differentiable, non- negative function defined...

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  3. Let y =f (x) be a twice differentiable, non- negative function defined...

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  4. Let y =f (x) be a twice differentiable, non- negative function defined...

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  5. The vlaue of ((5050)underset(0)overset(1)int(1-x^(50))^(100)dx)/(under...

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  6. lim(x->pi/4)(int2^(sec^2x)f(t) dt)/(x^2-(pi^2)/16) equal :

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  7. Match the integrals in Column I with the values in {:("Column II. ,...

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  8. Let S(n)=underset(k=1)overset(n)sum (n)/(n^(2)+nk+k^(2)) and T(n)=unde...

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  9. Let f be a non-negative function defined on the interval .[0,1].If int...

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  10. Let f:R to R be a function which satisfies f(x)=overset(x)underset(0...

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  11. The value of underset(xrarr0)(lim)(1)/(x^(3)) int(0)^(x)(tln(1+t))/(t^...

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  12. The value of overset(1)underset(0)int (x^(4)(1-x)^(4))/(1+x^(2))dx is

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  13. Let f be a real-valued function defined on the inverval (-1,1) such th...

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  14. For any real number x ,l e t[x] denote the largest integer less than o...

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  15. T h ev a l u eofint(sqrt(1n2))^(sqrt(1n3))(xsinx^2)/(sinx^2+sin(1n6-x^...

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  16. Let f:[1,oo] be a differentiable function such that f(1)=2. If int1...

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  17. The value of the integral int(-pi//2)^(pi//2) (x^(2) + log" (pi-x)/(pi...

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  18. Let f:[1/2,1]vecR (the set of all real numbers) be a positive, non-con...

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  19. Find a for which lim(n->oo) (1^a+2^a+3^a+...+n^a)/((n+1)^(a-1)[(na+1)+...

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  20. Let f:(0,oo)vecR be given by f(x)=int(1/x)^x(e^(-(t+1/t))dt)/t , then ...

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  21. Let f:[0,2]vecR be a function which is continuous on [0,2] and is diff...

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