∫ -5cos(πx) dx
Understand the Problem
The question is asking for the integral of the function -5cos(πx) with respect to x. To solve it, we will use integration techniques to find the antiderivative.
Answer
The integral is $ -\frac{5}{\pi} \sin(\pi x) + C $.
Answer for screen readers
The final result of the integral is:
$$ -\frac{5}{\pi} \sin(\pi x) + C $$
Steps to Solve
- Identify the integral to solve
We need to find the integral of the function $-5\cos(\pi x)$ with respect to $x$. The integral is expressed as:
$$ \int -5\cos(\pi x) , dx $$
- Use the integral rule for cosine
The integral of $\cos(kx)$ is $\frac{1}{k}\sin(kx)$, where $k$ is a constant. Here, $k = \pi$.
- Integrate the function
Applying the integral rule, we have:
$$ \int -5\cos(\pi x) , dx = -5 \cdot \frac{1}{\pi} \sin(\pi x) + C $$
Where $C$ is the constant of integration.
- Simplify the expression
Now, simplifying the expression yields:
$$ = -\frac{5}{\pi} \sin(\pi x) + C $$
The final result of the integral is:
$$ -\frac{5}{\pi} \sin(\pi x) + C $$
More Information
The function $-5\cos(\pi x)$ represents a scaled cosine wave. The antiderivative gives us a sine wave, and the factor $-\frac{5}{\pi}$ affects the amplitude.
Tips
- Forgetting to include the constant of integration, $C$, in the final answer.
- Incorrectly applying the integral rule for cosine, especially neglecting the factor $1/k$.
