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

  1. 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:

$$ F'(x) = \cos(x) $$

  1. 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.

  1. 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.

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