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

1. artofspeed Group Title

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

2. klimenkov Group Title

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

3. klimenkov Group Title

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

4. artofspeed Group Title

so it this whole expression correct?

5. sirm3d Group Title

only the first and last expressions can be equated.

6. artofspeed Group Title

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

7. klimenkov Group Title

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