∫ -5cos(πx) dx
Understand the Problem
The question is asking to evaluate the integral of the function -5cos(πx) with respect to x.
Answer
$$ -\frac{5}{\pi} \sin(\pi x) + C $$
Answer for screen readers
The final answer is:
$$ \int -5\cos(\pi x) , dx = -\frac{5}{\pi} \sin(\pi x) + C $$
Steps to Solve
- Identify the function to integrate
We need to evaluate the integral of the function $-5\cos(\pi x)$ with respect to $x$.
- Recall the integral formula for cosine
The integral of $\cos(kx)$ with respect to $x$ is given by: $$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$ where $C$ is the constant of integration.
- Apply the formula to our function
For our function, $-5\cos(\pi x)$, we have:
- $k = \pi$
- Thus, the integral becomes: $$ \int -5\cos(\pi x) , dx = -5 \cdot \frac{1}{\pi} \sin(\pi x) + C = -\frac{5}{\pi} \sin(\pi x) + C $$
- Write the final result
The evaluated integral is: $$ \int -5\cos(\pi x) , dx = -\frac{5}{\pi} \sin(\pi x) + C $$
The final answer is:
$$ \int -5\cos(\pi x) , dx = -\frac{5}{\pi} \sin(\pi x) + C $$
More Information
The integral of a cosine function results in a sine function. The inclusion of the coefficient $-5$ multiplied the result, and $\pi$ in the argument of cosine adjusts the amplitude accordingly. This adjustment reflects the frequency of the wave represented by the cosine function.
Tips
- Forgetting the constant of integration: It is important to always add the constant $C$ after computing an indefinite integral.
- Misapplying the integral formula: Ensure to use the correct form for cosine when integrating, especially remembering to account for the coefficient inside the function.
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