∫ - 5cos(πx) dx

Question image

Understand the Problem

The question is asking for the integral of the function -5cos(πx) with respect to x. We need to apply the rules of integration to solve this.

Answer

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

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

Steps to Solve

  1. Identify the function to integrate

We are given the integral of the function $-5\cos(\pi x)$.

  1. Apply the integral rule 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.

Here, $k = \pi$, so we will use that in our calculation.

  1. Perform the integration

Applying the rule we mentioned: [ \int -5\cos(\pi x) , dx = -5 \cdot \frac{1}{\pi} \sin(\pi x) + C ] This simplifies to: [ -\frac{5}{\pi} \sin(\pi x) + C ]

  1. State the final result

Thus, the final result of the integral is: [ -\frac{5}{\pi} \sin(\pi x) + C ]

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

More Information

The integral of cosine results in sine. The factor $-\frac{5}{\pi}$ arises from multiplying the constant $-5$ by $\frac{1}{\pi}$, which reflects the frequency of the cosine function.

Tips

  • Forgetting to include the constant of integration $C$ at the end of the result.
  • Confusing the integrals of sine and cosine; remember that $\int \cos(kx) dx$ results in a sine function.
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