Find the antiderivative of secant squared.
Understand the Problem
The question is asking for the antiderivative (or indefinite integral) of the function secant squared. This involves finding a function whose derivative is secant squared.
Answer
$\tan(x) + C$
Answer for screen readers
The antiderivative of $\sec^2(x)$ is $\tan(x) + C$.
Steps to Solve
- Identify the function to integrate
We are looking for the antiderivative of the function $\sec^2(x)$.
- Recall the derivative of a related function
We need to know a key relationship: the derivative of the tangent function is: $$ \frac{d}{dx}(\tan(x)) = \sec^2(x) $$
- Write the antiderivative
Since the derivative of $\tan(x)$ is equal to $\sec^2(x)$, we can conclude that the antiderivative is: $$ \int \sec^2(x) , dx = \tan(x) + C $$
Here, $C$ is the constant of integration.
The antiderivative of $\sec^2(x)$ is $\tan(x) + C$.
More Information
The function $\sec^2(x)$ is significant in calculus because it frequently appears in problems involving trigonometric integrals and differential equations. The antiderivative results in the tangent function, which is a fundamental function in trigonometry.
Tips
- Forgetting to include the constant of integration $C$ is a common mistake.
- Confusing $\sec^2(x)$ with other trigonometric functions and their derivatives.
