integral of sin(6x)
Understand the Problem
The question is asking for the integral of the function sin(6x), which involves calculating the antiderivative of this trigonometric function.
Answer
$$ -\frac{1}{6} \cos(6x) + C $$
Answer for screen readers
The final result of the integral is: $$ -\frac{1}{6} \cos(6x) + C $$
Steps to Solve
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Identify the Integral to Solve You are tasked with finding the integral of the function $\sin(6x)$. We express this as: $$ \int \sin(6x) , dx $$
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Use the Integral Formula for Sine The integral of $\sin(kx)$ is given by: $$ \int \sin(kx) , dx = -\frac{1}{k} \cos(kx) + C $$ where $C$ is the constant of integration and $k$ is a constant multiplier of $x$.
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Substitute the Value of k In our function, $k = 6$. According to the formula, we substitute $k$: $$ \int \sin(6x) , dx = -\frac{1}{6} \cos(6x) + C $$
The final result of the integral is: $$ -\frac{1}{6} \cos(6x) + C $$
More Information
This result shows that the antiderivative of $\sin(6x)$ is a function that transforms the sine function into a cosine function, with a coefficient that accounts for the frequency of the sine function. The constant $C$ represents any constant value since the process of integration can yield an infinite number of possible functions that differ only by a constant.
Tips
- Forgetting to divide by $k$ when using the integral formula for sine. Always remember to include this step.
- Not adding the constant of integration $C$ at the end of the integral. It's important to indicate that there are multiple antiderivatives differing by a constant.
