Question

find the antiderivative for each function when c equals 0. do as many as you can mentally. check your 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

Explanation:

Step1: Recall the antiderivative formula

The antiderivative of \(a\cos(bx)\) is \(\frac{a}{b}\sin(bx)+C\). Here \(a = 18\) and \(b=18\), and \(C = 0\).

Step2: Apply the formula

Substitute \(a = 18\) and \(b = 18\) into \(\frac{a}{b}\sin(bx)\). We get \(\frac{18}{18}\sin(18x)=\sin(18x)\).

Step3: Check by differentiation

Differentiate \(y=\sin(18x)\) using the chain rule. If \(y=\sin(u)\) where \(u = 18x\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Since \(\frac{dy}{du}=\cos(u)\) and \(\frac{du}{dx}=18\), \(\frac{dy}{dx}=18\cos(18x)\).

Answer:

\(\sin(18x)\)