## artofspeed Group Title integral: are these two expressions equal? one year ago one year ago

1. artofspeed

$\int\limits_{0}^{a}f(a-x)dx = \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx$

2. klimenkov

You can check this on example. Try $$f(x)=x^2, a=1.$$

3. klimenkov

It will more easy if you take $$f(x)=1$$.

4. artofspeed

so it this whole expression correct?

5. sirm3d

only the first and last expressions can be equated.

6. artofspeed

actually, the first and last aren't equal. But what's wrong in the expression?

7. klimenkov

$\int\limits_{0}^{a}f(a-x)dx \ne \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx$The primitive of $$f(a-x)$$ is $$-\int f(a-x)dx$$ and not $$\int f(a-x)dx$$. You can check this by taking derivative. Now, I think you can answer your question by yourself.