How to integrate tan(x)?

Understand the Problem

The question is asking for the process of integrating the function tan(x). We will approach it by using the integral of trigonometric functions.

Answer

$$ \int \tan(x) \, dx = -\ln|\cos(x)| + C $$ or $$ \int \tan(x) \, dx = \ln|\sec(x)| + C $$

Steps to Solve

Set up the integral

We want to find the integral of the function $\tan(x)$. This can be expressed as:
$$ \int \tan(x) , dx $$

Use the identity for tangent

Recall that $\tan(x)$ can be rewritten in terms of sine and cosine:
$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$

So we rewrite the integral:
$$ \int \tan(x) , dx = \int \frac{\sin(x)}{\cos(x)} , dx $$

Substitute for easier integration

Next, we can use a substitution. Let:
$$ u = \cos(x) $$
Then:
$$ du = -\sin(x) , dx $$
or:
$$ -du = \sin(x) , dx $$

Thus, the integral becomes:
$$ \int \tan(x) , dx = \int \frac{-du}{u} $$

Integrate the new expression

The integral of $\frac{-1}{u}$ is:
$$ -\ln|u| + C $$
where C is the constant of integration.

Substitute back in terms of x

Now we substitute $u$ back to express the result in terms of $x$:
$$ -\ln|\cos(x)| + C $$

Alternatively, we can express this using the absolute value:
$$ \ln|\sec(x)| + C $$

So, the final result for the integral is:
$$ \int \tan(x) , dx = -\ln|\cos(x)| + C $$
or
$$ \int \tan(x) , dx = \ln|\sec(x)| + C $$

$$ \int \tan(x) , dx = -\ln|\cos(x)| + C $$
or
$$ \int \tan(x) , dx = \ln|\sec(x)| + C $$

More Information

The integral of the tangent function is a common problem in calculus and is related to the properties of right triangles and the unit circle. The logarithmic form of the integral shows a deeper connection to exponential functions and growth rates.

Tips

Forgetting to use absolute values when dealing with logarithms can lead to incorrect answers.
Misplacing the negative sign during substitution is a common mistake. Always ensure to keep track of the negative sign when substituting with $du$.

Thank you for voting!