Problem 33 - Essential Calculus

Question

Find the indefinite integral. $$ \int \pi \sin \pi x d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\pi \sin(\pi x)\) with respect to x is \(-\cos(\pi x) + C\).

Step 1: Verify the Integral Function

Verify the function to be integrated: \(\int \pi \sin(\pi x) dx\). Notice that the integrand contains \(\sin(\pi x)\), so it's helpful to remember the integral of \(\sin(ax)\) is \(-(1/a) \cos(ax) + C\).

Step 2: Apply Constant Rule

Apply the constant rule in integration. The constant rule states that the integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function. Hence, we can move out the factor of \(\pi\) before integrating. The integrand becomes \(\pi \int \sin(\pi x) dx\).

Step 3: Apply the Integrals of Sin(ax)

Now we apply the rule that the integral of \(\sin(ax)\) is \(-(1/a)\cos(ax) + C\). In this case, \(a = \pi\). Hence, the indefinite integral is \(-(\frac{1}{\pi})\cos(\pi x) + C\).

Step 4: Reapply the Constant

In the previous step, \(\pi\) was factored out. Now it is time to reintroduce it, multiplying it back into our equation. The final solution becomes \(-\pi(\frac{1}{\pi})\cos(\pi x) + C = -\cos(\pi x) + C\).