∫ -5cos(πx) dx
Understand the Problem
The question is asking to find the integral of the function -5cos(πx) with respect to x, which requires using integration techniques.
Answer
The integral is $-\frac{5}{\pi} \sin(\pi x) + C$.
Answer for screen readers
The integral of the function is
$$ -\frac{5}{\pi} \sin(\pi x) + C $$
Steps to Solve
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Identify the integral
We are tasked with finding the integral of the function $-5\cos(\pi x)$ with respect to $x$. -
Apply the integration rule for cosine
The integral of $\cos(kx)$ with respect to $x$ is given by the formula:
$$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$
where $C$ is the constant of integration.
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Substitute the value of $k$
In our case, $k = \pi$. Thus, when integrating $-5\cos(\pi x)$, we have:
$$ \int -5\cos(\pi x) , dx = -5 \left( \frac{1}{\pi} \sin(\pi x) \right) + C $$
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Simplify the expression
Now, we can simplify it:
$$ = -\frac{5}{\pi} \sin(\pi x) + C $$
The integral of the function is
$$ -\frac{5}{\pi} \sin(\pi x) + C $$
More Information
The function $-5\cos(\pi x)$ represents a cosine wave scaled by -5. The integration process gives us a sine wave due to the integral of cosine. The constant $C$ represents the family of antiderivatives.
Tips
- A common mistake is forgetting to include the constant of integration $C$. Always remember that indefinite integrals have an arbitrary constant.
- Another mistake could be misapplying the rule for cosine integration, such as not correctly adjusting for the coefficient of $x$.
