# 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 = $\displaystyle \frac{(-ln x -2)}{x^2}$

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

i tried using substitution
$\displaystyle u = lnx + 3$
$\displaystyle h(u) = \frac{1}{x} u$
$\displaystyle = \frac{1}{x} (lnx + 3)$

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

5. perfect, thanks!