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 ( -8 cos (x) ).
c. the antiderivative of ( sin (11 x)-8 sin (8 x) ) is

Explanation:

Step1: Integrate \(\sin(11x)\)

Use the formula \(\int\sin(ax)dx =-\frac{1}{a}\cos(ax)+C\). For \(a = 11\), \(\int\sin(11x)dx=-\frac{1}{11}\cos(11x)\)

Step2: Integrate \(-8\sin(8x)\)

For \(a = 8\), \(\int-8\sin(8x)dx=-8\times(-\frac{1}{8})\cos(8x)=\cos(8x)\)

Step3: Combine the results

\(\int(\sin(11x)-8\sin(8x))dx=\int\sin(11x)dx-\int8\sin(8x)dx=-\frac{1}{11}\cos(11x)+\cos(8x)\)

Answer:

\(-\frac{1}{11}\cos(11x)+\cos(8x)\)