Integral. Why it's negative?

Question

I have an integral:

$$ \int \frac{5}{50-x} dx$$

Why it is equal to $$ -5 \ln|50-x| +constant $$?

I'm getting $$ 5 \ln|50-x| +constant $$

Answer 1

If you take the derivative of $5ln|50-x| + C$ What do you get?

Answer 2

$\dfrac{d}{dx}(50-x) = -1$.

$$\int \frac{5}{50-x} dx = -5\int \dfrac{(50 - x)'\,dx}{50 - x} = -5\ln|50 - x| + C$$

That is, if $u = 50-x$, then $du = -dx \iff dx = -\,du$

And our integral becomes $$5\int\dfrac{-du}{u}\quad =\quad -5\int \dfrac{du}{u} \quad = \quad -5\ln|u|+C = -5\ln|50 - x| + C$$

Answer 3

Let $u=50-x$, then $du=-dx$ which is the same as $-du=dx$. This makes the integral $$5\int\dfrac{1}{u}(-du)=-\int 5\dfrac{1}{u}du=-5\ln|u|+C=-5\ln|50-x|+C$$

Answer 4

set u=50-x and solve the integral.