∫ -5cos(πx) dx

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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

  1. 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:

$$ \int -5\cos(\pi x) , dx $$

  1. Use the integral rule for cosine

The integral of $\cos(kx)$ is $\frac{1}{k}\sin(kx)$, where $k$ is a constant. Here, $k = \pi$.

  1. 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.

  1. 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 $1/k$.

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