Finding the integral using u-substitution? If the integral from 3 to 5 of f(x-a)dx = 7 where a is a constant, then what it the integral from (3-a) to (5-a) of f(x)dx? Explain your thought process...

Finding the integral using u-substitution? If the integral from 3 to 5 of f(x-a)dx = 7 where a is a constant, then what it the integral from (3-a) to (5-a) of f(x)dx? Explain your thought process please. Thank you.

Asked on by modesthief

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lfryerda | High School Teacher | (Level 2) Educator

Posted on

The given information is `int_3^5 f(x-a)dx=7` .

Now, start with the question,

`int_{3-a}^{5-a}f(x)dx`   make the substitution `u=x+a` , so `x=u-a` and `du=dx` and the limits become `u=5-a+a=5` and `u=3-a+a=3` .  The integral is now:

`int_3^5f(u-a)du`   But since the variable of integration can be anything, this is equivalent to 

`int_3^5f(x-a)dx`   which is equal to 7.

The integral equals 7.

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