The integral of $\tan x$

Question

I am a very slow learning person in math. I am trying to learn integration but I can't find the best way to start solving an integral.

Now I am trying to solve

$$\int \tan x\, dx$$ but I dont know how I should start! Can someone provide any help?

Answer 1

$$ \int {\sin x \over \cos x}\,\mathrm dx$$

Now you can use this substitution:

$$u=\cos x \quad \mathrm du=-\sin x\,\mathrm dx$$

Answer 2

HINT: $\tan x =\dfrac{ \sin x}{ \cos x}$ and $(\cos x)'= - \sin x$.

Answer 3

with the last hint, consider $d(\cos x)=-\sin xdx$

Answer 4

$$\int \tan x\, dx=$$

$$\int \frac {\sin x }{\cos x } dx =$$

$$ \int \frac {-du}{u } =$$

$$\ln |sec (x)| +C$$