Integration of tan x
Understand the Problem
The question is asking for the integral of the function tan x. This involves finding the antiderivative of tan x, which is a process that requires an understanding of integral calculus.
Answer
$\ln\cos x + C$
Answer for screen readers
The integral of tan x is lncos x + C, where C is the constant of integration.
Steps to Solve

Recall integration by parts
We can use the integration by parts formula, but a more direct method is to use a trigonometric identity. Recall that $\tan x = \frac{\sin x}{\cos x}$.

Rewrite the integral
Rewrite the integral using this identity:
$$\int \tan x dx = \int \frac{\sin x}{\cos x} dx$$

Substitute u = cos x
Make the substitution $u = \cos x$. Then $du = \sin x dx$, or $du = \sin x dx$.
This transforms our integral to:
$$\int \frac{1}{u} du$$

Integrate
The integral of $\frac{1}{u}$ is $\lnu$. So we have:
$$\int \frac{1}{u} du = \lnu + C$$

Substitute back
Now substitute back $u = \cos x$:
$$\lnu + C = \ln\cos x + C$$
This is our final answer.
The integral of tan x is lncos x + C, where C is the constant of integration.
More Information
The natural logarithm in the answer arises from the integration of 1/x. The absolute value is necessary because the cosine function can be negative, and we can't take the logarithm of a negative number. The constant C represents the fact that differentiation eliminates constants, so when we integrate, we need to account for all possible constants.
Tips
A common mistake is to forget the negative sign or the absolute value bars in the answer. Remember that when you substitute u = cos x, you get a negative sign from du = sin x dx. Also, don't forget to include the constant of integration C.