# Thread: antiderivative with natural logs

1. ## antiderivative with natural logs

my prof posted the answers and his is different than mine, any step by step help to show what i'm doing wrong?

his answer = $\frac{(-ln x -2)}{x^2}$

"find the antiderivative":
$
f(x) = \frac{1}{x} (lnx + 3)$

i tried using substitution
$
u = lnx + 3$

$h(u) = \frac{1}{x} u$
$
= \frac{1}{x} (lnx + 3)$

.....stuck.

2. The professor made a mistake; that's the derivative, not the antiderivative.

3. can you explain to me the process of finding that antiderivative?

a little confused on how to go about it.

4. To find the antiderivative we have to perform an indefinite integral for $\frac{1}{x}(\ln x +3)$. For this particular function, it's best to use u-substitution. Let $u = \ln x + 3$. Then $du = \frac{dx}{x}$, so $\int \frac{1}{x}(\ln x + 3)\,dx = \int u\, du = \frac{u^{2}}{2} + C$. Now substituting our expression for $u$, we see that the antiderivative is $\frac{1}{2}(\ln x + 3)^{2} + C$.

5. perfect, thanks!