What is the integration of cos(3x)?
Understand the Problem
The question is asking for the integral of the function cos(3x), which requires applying integration rules. The integral will yield a function along with a constant of integration.
Answer
$$ \frac{1}{3} \sin(3x) + C $$
Answer for screen readers
The integral of $\cos(3x)$ is
$$ \frac{1}{3} \sin(3x) + C $$
Steps to Solve
- Identify the Integral to Solve
We need to solve the integral of the function $\cos(3x)$. This can be represented as:
$$ \int \cos(3x) , dx $$
- Use the Substitution Method
We will use the substitution method by letting $u = 3x$. Then, we find the differential $du$:
$$ du = 3 , dx $$
This means that:
$$ dx = \frac{du}{3} $$
- Rewrite the Integral
Now, substituting $u$ and $dx$ into the integral gives:
$$ \int \cos(3x) , dx = \int \cos(u) \cdot \frac{du}{3} $$
This simplifies to:
$$ \frac{1}{3} \int \cos(u) , du $$
- Integrate the Cosine Function
Now we can solve the integral of $\cos(u)$:
$$ \int \cos(u) , du = \sin(u) $$
So we have:
$$ \frac{1}{3} \sin(u) $$
- Back Substitute for $u$
We need to substitute back the original variable $x$ into our result:
$$ \frac{1}{3} \sin(3x) + C $$
where $C$ is the constant of integration.
The integral of $\cos(3x)$ is
$$ \frac{1}{3} \sin(3x) + C $$
More Information
The integration of trigonometric functions is a fundamental skill in calculus. The constant $C$ represents any constant value, since the derivative of a constant is zero.
Tips
- Forgetting to include the constant of integration $C$ after solving the integral.
- Incorrectly substituting back the value for $u$ or forgetting the factor from substitution.
