Find an antiderivative of the given function.
-$\cos \pi x + 6 \sin \frac { x } { 6 }$
A) $\frac { 1 } { \pi } \sin \pi x - \cos \frac { x } { 6 }$
B) $- \pi \sin \pi x + \cos \frac { x } { 6 }$
C) $- \sin \pi x - 36 \cos \frac { x } { 6 }$
D) $\frac { 1 } { \pi } \sin \pi x - 36 \cos \frac { x } { 6 }$

Question

Quizplus · Accepted Answer

We can use the fact that the antiderivative of $\cos(ax)$ is $\frac{1}{a}\sin(ax)$ and the antiderivative of $\sin(ax)$ is $-\frac{1}{a}\cos(ax)$.

Using this, we get:

$$\int \cos \pi x + 6 \sin \frac{x}{6} dx = \frac{1}{\pi} \sin \pi x - 6 \cos \frac{x}{6}+ C$$

However, we need to be careful with the factor of 6. To get rid of it, we can use the fact that $\cos \frac{x}{6} = \cos \left(\frac{1}{6} xight)$ and $\frac{1}{\pi} = \frac{6}{6\pi}$. This gives:

$$\int \cos \pi x + 6 \sin \frac{x}{6} dx = \frac{1}{\pi} \sin \pi x - 6 \cos \frac{x}{6}+ C = \frac{1}{6\pi} \sin \pi x - \cos \frac{x}{6} + C$$

Thus, the correct answer is $\boxed{	ext{(D)}}$.

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Find an Antiderivative of the Given Function $\cos \pi x + 6 \sin \frac { x } { 6 }$