Let  f:R\rightarrow R  be a function such that  

f(2-x)=f(2+x)\, and\, f(4-x)=f(4+x),for\, all\, x\, \epsilon\, R\, and\, \int ^{2}_{0}f(x)dx=5.\, Then\, the\, value\, of\, \int_{10}^{50}f(x)dx\: is:

  • Option 1)

    80

  • Option 2)

    100

  • Option 3)

    125

  • Option 4)

    200

 

Answers (1)

As learnt in concept

Properties of Definite Integration -

For periodic function

Let Period (T) then

\int_{0}^{nT}f(n)dx= n\int_{0}^{T}f(x)dx

 

- wherein

Where f(x) is periodic function with period T and n is any integer.

 

 \int_{0}^{2} f(x)dx =5

Also, \: f(2-x)= f(2+x)\: and \: f(4-x) =f(4-x)

On\: solving \: we \: get\: that\: period\: of \: f(x)\: is \: 2

Hence \int_{10}^{50}f(x)dx= 20\int_{0}^{2} f(x)dx

= 20\times 5 = 100


Option 1)

80

Incorrect

Option 2)

100

Correct

Option 3)

125

Incorrect

Option 4)

200

Incorrect

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