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If f(t) is an odd function, then varphi(...

If `f(t)` is an odd function, then `varphi(x)=int_a^xf(t)dx` is an even function.

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If f(t) is an odd function,then varphi(x)=int_(a)^(x)f(t)dt is an even function.

Statement 1: If f(x) is an odd function,then f'(x) is an even function.Statement 2: If f'(x) is an even function,then f(x) is an odd function.

STATEMENT 1:int_(a)^(x)f(t)dt is an even function if f(x) is an odd function.STATEMENT 2:int_(a)^(x)f(t)dx is an odd function if f(x) is an even function.

Statement I The function f(x) = int_(0)^(x) sqrt(1+t^(2) dt ) is an odd function and g(x)=f'(x) is an even function , then f(x) is an odd function.

If f(t) is a continuous function defined on [a,b] such that f(t) is an odd function, then the function phi(x)=int_(a)^(x) f(t)dt

Given f an odd function periodic with period 2 continuous AA x in R and g(x)=int_0^x f(t)dt then (i) g(x) is an odd function (ii) g(x+2)=1 (iii) g(2)=0 (iv) g(x) is an even function

If f(x) is an odd function,then write whether f'(x) is even or odd.

Prove that the derivative of an even function is an odd function and that of an odd function is an even function.

If f(x) is an even function , then what is int_0^pi f (cos x) dx equal to

A DAS GUPTA-Properties and Application of definite Integrals-EXERCISE
  1. Show that: inta^bf(x)dx=int(a+c)^(b+c)f(x-c)dx and hence show that int...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  18. 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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  19. The value of int0^(pi/2)sin2x.log tanxdx is

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

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