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Question

Let f(x)=x0g(t)dt, where g is a non-zero even function. If f(x+5)=g(x), then x0f(t)dt equals :

A
5x+5g(t)dt
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B
x+55g(t)dt
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C
2x+55g(t)dt
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D
55x+5g(t)dt
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Solution

The correct option is A 5x+5g(t)dt
f(x)=x0g(t)dt
f(x)=g(x)
Given, g is even f is even.
f is odd.

f(x+5)=g(x)
f(x)=g(x5) (1)
Now, consider I=x0f(t)dt
Put t=u5dt=du
I=x+55f(u5)du
I=x+55f((5u))du
I=x+55f(5u)du
I=x+55g(u)du [From (1)]
I=x+55g(u)du
I=5x+5g(t)dt



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