# Math Help - Which one the right integration

1. ## Which one the right integration

I've a question that is bothering me for couple of days now. i can't find a valid answer. Which one is correct below?

$
x \int \frac{1}{x} \mathrm dx = x \ln\left|x\right| + C
$

or

$
x \int \frac{1}{x} \mathrm dx = x \ln\left|x\right| + Cx
$

Are they both correct? why?

2. The second one is correct, because the arbitrary constant is associated with the integral. Since the x on the outside multiplies the entire integral, it must also multiply the constant.

3. Originally Posted by Ackbeet
The second one is correct, because the arbitrary constant is associated with the integral. Since the x on the outside multiplies the entire integral, it must also multiply the constant.
Thanks. But if the second one is correct then we know that if two integrals are equal they are differed by a constant. so LHS and RHS are differed by a constant C. So shouldn't
$
x \int \frac{1}{x} \mathrm dx = x \ln\left|x\right| + C
$

be right also? Am i interpreting the theory right?

4. Because the variable $x$ is outside the integral that is why x is multiplied with the whole integral including the constant. Sorry for posting before reading the reply carefully. Thanks Ackbeet.

5. You're welcome. Have a good one!