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

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

A

is an odd function

B

is an even function

C

is an increasing function on [a,b]

D

none of these

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The correct Answer is:
B
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OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Exercise
  1. The value of int(0)^(2pi) cos^(99)x dx, is

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  2. If f(a+x)=f(x), then int(0)^(na) f(x)dx is equal to (n in N)

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  3. If f(t) is an odd function, then prove that varphi(x)=inta^xf(t)dt is ...

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  4. If f(x) is an integrable function over every interval on the real line...

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  5. If I(1)=int(3pi)^(0) f(cos^(2)x)dx and I(2)=int(pi)^(0) f(cos^(2)x) th...

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  6. If f(x) is a quadratic polynomial in x such that 6int0^1 f(x)dx-{f(0...

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  7. The value of integral int(-2)^(4) x[x]dx is

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  8. If h(a)=h(b), the value of the integral inta^b [f(g(h(x))]^(-1)f'(g...

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  9. Given that, F(x)=(1)/(x^(2))int(4)^(x)(4t^(2)-2F'(t))dt, find F'(4).

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

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  11. Suppose for every integer n, . underset(n)overset(n+1)intf(x)dx = n^(2...

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  12. int(-pi+4)^(pi//4) (tan^(2)x)/(1+a^(x))dx is equal to

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  13. The value of int(0)^(pi//2) cosec(x-pi//3)cosec(x-pi//6)dx is

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  14. The value of int(-1)^(1)(x|x|)dx is equal to

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  15. int0^3 |x^(3)+x^(2)+3x|dx is equal to

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  16. Evaluate: int(-1)^1(x^3+|x|+1)/(x^2+2|x|+1)dx

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  17. Evaluate: int(-pi/2)^(pi/2)log{(a x^2+b x+c)/(a x^2-b x+c)(a+b)|sinx|}...

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  18. For any natural number n, the value of the integral int(0)^(sqrt(n))...

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  19. For any n in R^(+), the value of the integral int(0)^(n[x]) (x-[x])d...

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  20. Let d/(dx)F(x)=((e^(sinx))/x),x > 0. If int1^4 3/x e^sin (x^3)dx=F(...

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