# 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.

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Expert Answers

lfryerda | Certified Educator

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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