Question

find the antiderivative for each function when c equals 0. do as many as you can mentally. check your answers by differentiation.
a. $-8\csc x\cot x$
b. $-\csc (6x)\cot (6x)$
c. $-\pi\csc \left(\frac{\pi x}{4}\
ight)\cot \left(\frac{\pi x}{4}\
ight)$
a. the antiderivative of $-8\csc x\cot x$ is

Explanation:

Step1: Recall the antiderivative formula

We know that the antiderivative of \(\csc x\cot x\) is \(-\csc x + C\).

Step2: Apply the constant - multiple rule

For the function \(y=-8\csc x\cot x\), using the rule \(\int kf(x)dx = k\int f(x)dx\) (where \(k = - 8\) and \(f(x)=\csc x\cot x\)).
Since \(\int\csc x\cot xdx=-\csc x + C\), then \(\int-8\csc x\cot xdx=-8\int\csc x\cot xdx\).
Substituting the antiderivative of \(\csc x\cot x\), we get \(\int-8\csc x\cot xdx=-8(-\csc x)+0 = 8\csc x\) (because \(C = 0\)).

Step3: Check by differentiation

Differentiate \(y = 8\csc x\) using the formula \(\frac{d}{dx}(\csc x)=-\csc x\cot x\).
By the constant - multiple rule \(\frac{d}{dx}(8\csc x)=8\frac{d}{dx}(\csc x)\).
Since \(\frac{d}{dx}(\csc x)=-\csc x\cot x\), then \(8\frac{d}{dx}(\csc x)=8(-\csc x\cot x)=-8\csc x\cot x\).

Answer:

\(8\csc x\)