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If f(x) is periodic function with period...

If f(x) is periodic function with period, T, then

A

`underset(a)overset(b)int f(x)dx=underset(a)overset(b+T)intf(x)dx`

B

`underset(a)overset(b)int f(x)dx=underset(a+T)overset(b)intf(x)dx`

C

`underset(a)overset(b)int f(x)dx=underset(a+T)overset(b+T)intf(x)dx`

D

`underset(a)overset(b)int f(x)dx=underset(a+T)overset(b+2T)intf(x)dx`

Text Solution

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The correct Answer is:
C
We have,
`underset(a+T)overset(b+T)int f(x)dx=underset(a)overset(b)intf(y+T)` dy, where `x=y+T`
`underset(a+T)overset(b+T)int f(x)dx=underset(a)overset(b)intf(x)dy[ :. "f is periodic with periodT" :. f(y+T)=f(y)]`
`underset(a+T)overset(b+T)int f(x)dx=underset(a)overset(b)intf(x)dx`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. If f(x) is an odd pefiodc function defined on the interval [T/2,T/2], ...

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  2. If int(pi//2)^(theta) sin x dx=sin 2 theta then the of theta satisfyin...

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  3. If f(x) is periodic function with period, T, then

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  4. If f(n)=(1)/(n){(n+1)(n+2)(n+3)...(n+n)}^(1//n) then lim(n to oo)f(n)...

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  5. The poins of extremum of phi(x)=int(1)^(x) e^(-t^(2)//2) (1-t^(2))dt a...

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  6. int(-2)^(2) min(x-[x],-x-[x])dx equals, where [x] represents greates i...

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  7. The integral int(0)^(a) (g(x))/(f(x)+f(a-x))dx vanishes, if

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  8. If (1)/(sqrt(a))int(1)^(a)((3)/(2)sqrt(x)+1-(1)/(sqrt(x)))dx lt4 then ...

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  9. The expression (int(0)^(n)[x]dx)/(int(0)^(n){x}dx)a where [x] and [x] ...

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  10. If f(x)= Min {|x-1|,|x|,|x+1|} "then" int(-1)^(1)f(x)dx eqals

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  11. The value of int(0)^(100) [ tan^(-1)x]dx is

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  12. int(-1)^(10) sgn (x-[x])dx equals

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  13. If n in N, then int(-n)^(n)(-1)^([x]) dx equals

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  14. The value of :.int(0)^([x]) (2^(x))/(2^([x]))dx is

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  15. If f(x) rArr int(sin theta)^("cosec" theta) f(x)dx equals

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  16. If f(x)={{:(e^(cosx), sin x, "for" |x|le2),(,2,"otherwise"):} then int...

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  17. The value of the integral int(e^(-1))^(e^(2)) |(log(e)x)/(x)|dx is

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  18. Let g(x)=int(0)^(x)f(t) dt, where f is such that (1)/(2)lef(t)le1 "fo...

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  19. Let f:(0, oo) in R and F(x) = int(0)^(x) f(t) dt. If F(x^(2))=x^(2)(1+...

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  20. The integral int( -1//2)^(1//2) {[x]+ in ((1+x)/(1-x))} dx equals

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