∫ - 5cos(πx) dx

Understand the Problem

The question involves performing an integral calculation of the function -5cos(πx). We need to find the antiderivative of this function with respect to x.

Answer

$$ -\frac{5}{\pi} \sin(\pi x) + C $$

Steps to Solve

Identify the function to integrate

We want to find the antiderivative of the function ( -5\cos(\pi x) ).

Apply the integral property for cosine

Recall that the integral of ( \cos(kx) ) with respect to ( x ) is given by: $$ \int \cos(kx) , dx = \frac{1}{k} \sin(kx) + C $$

In our case, ( k = \pi ).

Integrate the function

Now, apply the integral property: $$ \int -5\cos(\pi x) , dx = -5 \cdot \frac{1}{\pi} \sin(\pi x) + C $$

Simplify the expression

This simplifies to: $$ -\frac{5}{\pi} \sin(\pi x) + C $$

The antiderivative of the function is: $$ -\frac{5}{\pi} \sin(\pi x) + C $$

More Information

This integral represents the family of functions whose derivative is ( -5\cos(\pi x) ). The constant ( C ) represents any constant value, as the integral of a function can vary by a constant.

Tips

Forgetting to divide by the coefficient of ( x ) in the cosine function. Always remember to apply the $1/k$ factor when integrating ( \cos(kx) ).
Not including the constant of integration ( C ) in the final answer.

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