Question

evaluate the indefinite integral. (use ( c ) for the constant of integration.)
int \tan ^{8}(\theta) sec ^{2}(\theta) d \theta
illustrate and check that your answer is reasonable by graphing both the function and its antiderivative in the same viewing window (take ( c = 0 )).

Explanation:

Step1: Substitute \(u = \tan\theta\)

Since \(du=\sec^{2}\theta d\theta\), the integral \(\int\tan^{8}\theta\sec^{2}\theta d\theta\) becomes \(\int u^{8}du\).

Step2: Integrate \(u^{8}\)

Using the power - rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n = 8\), we have \(\int u^{8}du=\frac{u^{9}}{9}+C\).

Step3: Substitute back \(u=\tan\theta\)

We get \(\frac{\tan^{9}\theta}{9}+C\).

Answer:

\(\frac{\tan^{9}\theta}{9}+C\)