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Show that: inta^bf(x)dx=int(a+c)^(b+c)f(...

Show that: `int_a^bf(x)dx=int_(a+c)^(b+c)f(x-c)dx` and hence show that `int_0^pi sin^100xcos^99xdx=0`

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A DAS GUPTA-Properties and Application of definite Integrals-EXERCISE
  1. Evaluate the following:int0^(10pi)|cosx|dx

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

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

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

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

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

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

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

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

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

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

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

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

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

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  16. State whether the statements are true or false.int0^(pi/2)dx/(sqrt(sin...

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  17. If int0^a{f(x)+f(-x)}dx=int-a^aphi(x)dxthen phi(x)=.

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  18. Evaluate: int0^pi (cos x)/(1+sinx)^2 dx

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  19. The value int(-pi/4)^(pi/4)x^2sin^-1xdx=.

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  20. If int0^pi x f(sinx) dx=A int0^(pi/2) f(sinx)dx, then A is (A) pi/2 (B...

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