What is the antiderivative of tan(x)?
Understand the Problem
The question is asking for the antiderivative (or indefinite integral) of the function tan(x). This requires applying integral calculus techniques to find a function whose derivative is tan(x).
Answer
-\ln|\cos(x)| + C
Answer for screen readers
The final answer is -ln|cos(x)| + C
Steps to Solve
- Rewrite tan(x) using trigonometric identity
We know that $\tan(x) = \frac{\sin(x)}{\cos(x)}$. So, we can rewrite our integral as: $$ \int \tan(x) , dx = \int \frac{\sin(x)}{\cos(x)} , dx $$
- Use substitution method
Let $u = \cos(x)$. Then, $du = -\sin(x) , dx$, implying $-du = \sin(x) , dx$.
- Integrate in terms of u
Substitute $u$ and $du$ into the integral: $$ \int \frac{\sin(x)}{\cos(x)} , dx = \int \frac{-du}{u} = -\int \frac{1}{u} , du $$
- Find the antiderivative
The antiderivative of $\frac{1}{u}$ is $\ln|u|$. Therefore, we have: $$ -\int \frac{1}{u} , du = -\ln|u| + C $$
- Substitute back in terms of x
Recall that $u = \cos(x)$. Substitute back to get the final answer: $$ -\ln|\cos(x)| + C $$
The final answer is -ln|cos(x)| + C
More Information
The constant C represents the constant of integration which can be any real number.
Tips
A common mistake is forgetting to include the absolute value in the logarithm, or neglecting to add the constant of integration C.