Question

find the antiderivative for each function when c equals 0.
a. ( f(x)=-11 sin (11 x) ) b. ( g(x)=8 sin (x) ) c. ( h(x)=sin (11 x)-8 sin (8 x) )
a. the antiderivative of ( -11 sin (11 x) ) is ( cos (11 x) ).
b. the antiderivative of ( 8 sin (x) ) is ( square ).

Explanation:

Step1: Recall the antiderivative formula for \(\sin(x)\)

The antiderivative of \(\sin(x)\) is \(-\cos(x)+C\). For a function \(a\sin(bx)\), its antiderivative is \(-\frac{a}{b}\cos(bx)+C\) (by the chain - rule in reverse, \(\int a\sin(bx)dx\), let \(u = bx\), \(du=bdx\), \(dx=\frac{1}{b}du\), \(\int a\sin(bx)dx=\frac{a}{b}\int\sin(u)du=-\frac{a}{b}\cos(u)+C = -\frac{a}{b}\cos(bx)+C\)).

Step2: Apply the formula to \(g(x)=8\sin(x)\)

Here \(a = 8\) and \(b = 1\). Using the formula \(\int a\sin(bx)dx=-\frac{a}{b}\cos(bx)+C\), when \(C = 0\), we have \(\int8\sin(x)dx=- 8\cos(x)+0\).

Answer:

\(-8\cos(x)\)