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. -π csc (πx/4) cot (πx/4)
a. the antiderivative of -8 csc x cot x is 8 csc x.
b. the antiderivative of - csc (6x) cot (6x) is

Explanation:

Step1: Recall the antiderivative formula

We know that the antiderivative of \( \csc u\cot u\) with respect to \(u\) is \(-\csc u + C\). For the function \(y =-\csc(6x)\cot(6x)\), let \(u = 6x\), then \(du=6dx\), and \(dx=\frac{1}{6}du\).

Step2: Apply the substitution

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Using the formula \(\int\csc u\cot udu =-\csc u + C\), we have \(\frac{1}{6}\int-\csc(u)\cot(u)du=\frac{1}{6}\csc(u)+C\). Substituting back \(u = 6x\), we get \(\frac{1}{6}\csc(6x)+C\). Since \(C = 0\), the antiderivative is \(\frac{1}{6}\csc(6x)\).

Step3: Check by differentiation

Differentiate \(y=\frac{1}{6}\csc(6x)\) using the chain rule. The derivative of \(\csc t\) is \(-\csc t\cot t\). Let \(t = 6x\), then \(y'=\frac{1}{6}(-\csc(6x)\cot(6x))\times6=-\csc(6x)\cot(6x)\)

Answer:

\(\frac{1}{6}\csc(6x)\)