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Show that: int0^x[x]dx=[x]([x]-1)/2+[x](...

Show that: `int_0^x[x]dx=[x]([x]-1)/2+[x](x-[x])`, where `[x]` denotes the integral part of `x`.

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A DAS GUPTA-Properties and Application of definite Integrals-EXERCISE
  1. Prove that inta^b{[x]+[-x]}dx=a-b,where [x]is the greatest integer lex...

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  2. Statement-I int0^9[sqrtx]dx=13, Statement-II int0^(n^2) [sqrt x]dx=(n(...

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  3. Show that: int0^x[x]dx=[x]([x]-1)/2+[x](x-[x]), where [x] denotes the ...

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  4. The value of the integral overset(100pi)underset(0)int sqrt(1-cos2x)" ...

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  5. Evaluate the following : int(0)^(pi)(dx)/(1+sinx)

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  6. Evaluate the following:int0^(10pi)|cosx|dx

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  7. Evaluate int(0)^(npi+t)(|cosx|+|sinx|)dx, where n epsilonN and t epsil...

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  8. Show that: inta^bf(x)dx=int(a+c)^(b+c)f(x-c)dx and hence show that int...

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  9. If f(x) is a continuous function defined on [0,\ 2a]dot\ Then prove t...

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  10. If f(t) is an odd function, then varphi(x)=inta^xf(t)dx is an even fun...

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  11. It is known that f(x) is an odd function in the interval [p/2, p/2] an...

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  12. If f(a+x)=f(x), then prove that inta^(n a)f(x)dx=(n-1)int0^af(x)dx whe...

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  13. Evaluate: int0^(pi/2) (sin2x)/(sin^4x+cos^4x)dx

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  14. Let a, b, c be non-zero real numbers such that ; int0^1 (1 + cos^8 x)(...

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  15. If f(X) is continuous and ninNthen the value of int-2^2{f(x)-f(-x)}x^...

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  16. An extremum value of y=int0^x(t-1)(t-2)dt is : 5/6 (b) 2/3 (c) 1 ...

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  17. An AC voltage is given by E=E(0) sin(2pit)/(T). Then the mean value of...

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  18. If g(x)=int0^xcos^4tdt , then g(x+pi) equals g(x)+g(pi) (b) g(x)-g(pi...

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  19. The value of int(-pi)^(pi)(2x(1+sinx))/(1+cos^(2)x)dx is

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  20. State whether the statements are true or false.If y=int0^xf(t)dt .Then...

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