∫ -5cos(πx) dx
Understand the Problem
The question is asking to solve the integral of -5cos(πx) with respect to x. The approach involves applying integration techniques to find the antiderivative of the given function.
Answer
The integral is $-\frac{5}{\pi} \sin(\pi x) + C$.
Answer for screen readers
The final answer is: $$ -\frac{5}{\pi} \sin(\pi x) + C $$
Steps to Solve
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Identify the integral We need to find the integral: $$ \int -5 \cos(\pi x) , dx $$
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Factor out constant Since $-5$ is a constant, we can factor it out of the integral: $$ -5 \int \cos(\pi x) , dx $$
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Apply integration formula The integral of $\cos(kx)$ is given by: $$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$ In our case, ( k = \pi ), so: $$ \int \cos(\pi x) , dx = \frac{1}{\pi} \sin(\pi x) + C $$
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Combine results Substituting this back into our integral gives: $$ -5 \left( \frac{1}{\pi} \sin(\pi x) + C \right) $$ This results in: $$ -\frac{5}{\pi} \sin(\pi x) - 5C $$
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Simplify the constant term We can rename the constant term, reflecting it as: $$ -\frac{5}{\pi} \sin(\pi x) + C' $$ where ( C' = -5C ) is still a constant.
The final answer is: $$ -\frac{5}{\pi} \sin(\pi x) + C $$
More Information
This result represents the indefinite integral of the function, giving us the antiderivative along with a constant of integration ( C ), as is customary in indefinite integrals.
Tips
- Forgetting to factor out the constant before integrating: Always factor out constants to simplify integration.
- Not applying the integration formula correctly for trigonometric functions: Always remember that $\int \cos(kx) , dx = \frac{1}{k} \sin(kx)$.
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