∫ -5cos(πx) dx

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Understand the Problem

The question is asking us to find the integral of the function -5cos(πx) with respect to x. This requires knowledge of integration techniques to solve the problem.

Answer

$$ -\frac{5}{\pi} \sin(\pi x) + C $$
Answer for screen readers

The final answer is: $$ -\frac{5}{\pi} \sin(\pi x) + C $$

Steps to Solve

  1. Set Up the Integral

We begin with the integral we need to solve: $$ \int -5\cos(\pi x) , dx $$

  1. Factor Out the Constant

We can factor the constant (-5) out of the integral: $$ -5 \int \cos(\pi x) , dx $$

  1. Integrate Cosine Function

The integral of $\cos(kx)$ is $\frac{1}{k} \sin(kx) + C$. Here, $k = \pi$: $$ -5 \cdot \frac{1}{\pi} \sin(\pi x) + C $$

  1. Simplify the Expression

Now simplify this expression: $$ -\frac{5}{\pi} \sin(\pi x) + C $$

The final answer is: $$ -\frac{5}{\pi} \sin(\pi x) + C $$

More Information

This integral is a common example in calculus, demonstrating the use of standard integration techniques. The function $\cos(\pi x)$ has a periodic nature, which is reflected in the sine function that results from the integration. The constant $C$ represents the constant of integration.

Tips

  • Confusing the integral of cosine with that of sine: Remember that the integral of $\cos(kx)$ leads to $\frac{1}{k} \sin(kx)$.
  • Forgetting to include the constant of integration $C$ at the end.
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