integral of cos(6x)
Understand the Problem
The question is asking for the integral of the function cos(6x), which requires applying integration techniques. This involves finding the antiderivative of the cosine function multiplied by a constant.
Answer
$$ \int \cos(6x) \, dx = \frac{1}{6} \sin(6x) + C $$
Answer for screen readers
$$ \int \cos(6x) , dx = \frac{1}{6} \sin(6x) + C $$
Steps to Solve
- Identifying the integral to solve
We are asked to find the integral of the function $\cos(6x)$. This can be written as:
$$ \int \cos(6x) , dx $$
- Applying the integration rule for cosine
We use the fact that the integral of $\cos(kx)$ is given by:
$$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$
where $C$ is the constant of integration, and $k$ is a constant. Here, $k = 6$.
- Substituting the value of k
Now, we substitute $k = 6$ into the integration formula:
$$ \int \cos(6x) , dx = \frac{1}{6} \sin(6x) + C $$
- Final form of the integral
Thus, the final expression for the integral of $\cos(6x)$ is:
$$ \int \cos(6x) , dx = \frac{1}{6} \sin(6x) + C $$
$$ \int \cos(6x) , dx = \frac{1}{6} \sin(6x) + C $$
More Information
The integral of cosine functions is a common operation in calculus. The factor of $1/6$ comes from the chain rule used in reverse during integration. This is related to the scaling effect of the constant multiplier in the argument of the cosine function.
Tips
- Forgetting to include the constant of integration $C$. Always remember to add it when finding indefinite integrals.
- Misapplying the formula for the integral of $\cos(kx)$. Ensure that the correct formula and substitution are used.
