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Chapter 4: Problem 33

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

Short Answer

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

Step by step solution

01

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 \(-(1/a) \cos(ax) + C\).
02

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 \(\pi \int \sin(\pi x) dx\).
03

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 \(-(\frac{1}{\pi})\cos(\pi x) + C\).
04

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 \(-\pi(\frac{1}{\pi})\cos(\pi x) + C = -\cos(\pi x) + C\).

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