What is the antiderivative of cos?
Understand the Problem
The question is asking for the antiderivative of the cosine function, which is a fundamental concept in calculus. To solve this, we will identify the function whose derivative is cosine.
Answer
$$ \int \cos(x) \, dx = \sin(x) + C $$
Answer for screen readers
$$ \int \cos(x) , dx = \sin(x) + C $$
Steps to Solve
- Identify the Function
We need to find a function, $F(x)$, such that its derivative equals $\cos(x)$. This leads us to set up the equation:
$$ F'(x) = \cos(x) $$
- Recall the Antiderivative of Cosine
From differentiation rules, we know that the derivative of $\sin(x)$ is $\cos(x)$. Therefore, we conclude that the antiderivative of $\cos(x)$ is:
$$ F(x) = \sin(x) + C $$
where $C$ is the constant of integration.
- Conclude with the Result
Thus, we express the complete solution for the antiderivative:
$$ \int \cos(x) , dx = \sin(x) + C $$
$$ \int \cos(x) , dx = \sin(x) + C $$
More Information
The function $\sin(x)$ is integral to many areas in mathematics and physics. The constant $C$ represents an infinite number of functions that differ by a constant, all having the same derivative.
Tips
- Forgetting to include the constant of integration $C$ when determining the antiderivative.
- Confusing $\sin(x)$ with $\cos(x)$ in terms of their derivatives.