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If f(a+x)=f(x), then prove that inta^(n ...

If `f(a+x)=f(x),` then prove that `int_a^(n a)f(x)dx=(n-1)int_0^af(x)dx` where `a >0 and n in N.`

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A DAS GUPTA-Properties and Application of definite Integrals-EXERCISE
  1. If f(t) is an odd function, then varphi(x)=inta^xf(t)dx is an even fun...

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

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

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

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

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

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

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

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

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

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

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

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

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

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  16. 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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  17. The value of int0^(pi/2)sin2x.log tanxdx is

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  18. If int0^1e^t/(t+1) dt=a, then int(b-1)^b e^(-t)/(t-b-1) dt=

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  19. Evaluateint(-pi/2)^(pi/2)(sin2x)/(1+cos^3x)dx

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  20. If f is a continuous function on the interval [a,b] and there exists ...

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