What is the integral of tan x dx?
Understand the Problem
The question is asking for the integral of the function tan(x) with respect to x. The approach to solve this involves using the identity that relates tan(x) to sin(x) and cos(x) and applying integration techniques.
Answer
$\ln |\sec(x)| + C$
Answer for screen readers
The integral of $\tan(x)$ with respect to $x$ is $\ln |\sec(x)| + C$.
Steps to Solve
- Express tan(x) in terms of sin(x) and cos(x)
We know that the tangent function can be written as: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
- Identify the substitution
To integrate $\tan(x)$, we can use the substitution method. Let: $$ u = \cos(x) $$ Then, we differentiate: $$ du = -\sin(x) , dx \implies dx = -\frac{1}{\sin(x)} du $$
- Rewrite the integral
Using our substitution, the integral becomes: $$ \int \tan(x) , dx = \int \frac{\sin(x)}{u} \left(-\frac{1}{\sin(x)}\right) du = -\int \frac{1}{u} du $$
- Integrate
Now we can integrate: $$ -\int \frac{1}{u} , du = -\ln |u| + C $$
- Substitute back in terms of x
Remember to substitute $u$ back to its original variable: $$ -\ln |\cos(x)| + C $$
- Final expression
We can also use properties of logarithms: $$ \int \tan(x) , dx = \ln |\sec(x)| + C $$
The integral of $\tan(x)$ with respect to $x$ is $\ln |\sec(x)| + C$.
More Information
This integral result is useful in calculus, particularly when dealing with trigonometric functions. The secant function, $\sec(x)$, is defined as $\sec(x) = \frac{1}{\cos(x)}$, which leads us back to our original functions.
Tips
- Forgetting to include the absolute value in the logarithm. Always use $|\ldots|$ when working with logarithms of trigonometric functions to handle negative values correctly.
- Not applying the substitution properly, which can lead to incorrect integrals.
