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Question

If f is a continuous function then,  2a0f(x)dx=a0f(x)dx+2a0f(2ax)dx.
 


Your Answer
A
True
Correct Answer
B
False

Solution

The correct option is B False
Let us try to see using the properties of definite integrals whether we are able to reach at the given equation.
LHS=2a0f(x)dx
=a0f(x)dx+2aaf(x)dx  (using property)                           ….(1)
Since it’s (2a - x) instead of (x) on the RHS we will try to make a substitution,
x = 2a - t
i.e., t = 2a - x
dx = -dt
 Lower limit = 2a - a
                        = a.
Upper limit = 2a - 2a
                            = 0
2aaf(x)dx=0af(2at)(dt)
=a0f(2at)dt  (using property)
=a0f(2ax)dx   (using property)    (2)
From (1) and (2) 
LHS=  a0f(x)dx+a0f(2ax)dx
 RHS
  The given statement is false.


 

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