Question

find the antiderivative for each function when c equals 0. do as many as you can mentally. c answers by differentiation.
a. ( 18 cos 18 x ) b. ( \frac{3 pi}{2} cos \frac{3 pi x}{2} ) c. ( cos \frac{9 pi x}{2}+10 pi cos x )
a. the antiderivative of ( 18 cos 18 x ) is ( sin (18 x) ).
b. the antiderivative of ( \frac{3 pi}{2} cos \frac{3 pi x}{2} ) is ( sin left(\frac{3 pi x}{2}
ight) ).
(type an exact answer, using ( pi ) as needed.)
c. the antiderivative of ( cos \frac{9 pi x}{2}+10 pi cos x ) is
(type an exact answer, using ( pi ) as needed.)

Explanation:

Step1: Find antiderivative of \(\cos\frac{9\pi x}{2}\)

Recall \(\int\cos(ax)dx=\frac{1}{a}\sin(ax)+C\). For \(y = \cos\frac{9\pi x}{2}\), \(a=\frac{9\pi}{2}\). So \(\int\cos\frac{9\pi x}{2}dx=\frac{2}{9\pi}\sin\frac{9\pi x}{2}\)

Step2: Find antiderivative of \(10\pi\cos x\)

Recall \(\int\cos xdx=\sin x + C\). For \(y = 10\pi\cos x\), \(\int10\pi\cos xdx=10\pi\sin x\)

Step3: Combine the results

By \(\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx\), \(\int(\cos\frac{9\pi x}{2}+10\pi\cos x)dx=\frac{2}{9\pi}\sin\frac{9\pi x}{2}+10\pi\sin x\)

Answer:

\(\frac{2}{9\pi}\sin\frac{9\pi x}{2}+10\pi\sin x\)