∫ - 5cos(πx) dx
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
- Identify the function to integrate
We are given the integral of the function $-5\cos(\pi x)$.
- 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.
- 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 ]
- 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.
