Question

evaluate the indefinite integral. (use c for the constant of integration.)

$$ int \frac { d t } { cos ^ { 2 } ( t ) sqrt 6 { 7 + \tan ( t ) } } $$

Explanation:

Step1: Use substitution

Let \(u = 7+\tan(t)\), then \(du=\sec^{2}(t)dt=\frac{dt}{\cos^{2}(t)}\).

Step2: Rewrite the integral

The integral \(\int\frac{dt}{\cos^{2}(t)\sqrt[6]{7 + \tan(t)}}\) becomes \(\int\frac{du}{u^{\frac{1}{6}}}\).

Step3: Integrate using the power rule

The power - rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(n=-\frac{1}{6}\), we have \(\int u^{-\frac{1}{6}}du=\frac{u^{-\frac{1}{6}+1}}{-\frac{1}{6}+1}+C=\frac{u^{\frac{5}{6}}}{\frac{5}{6}}+C=\frac{6}{5}u^{\frac{5}{6}}+C\).

Step4: Substitute back

Substitute \(u = 7+\tan(t)\) back into the expression.

Answer:

\(\frac{6}{5}(7+\tan(t))^{\frac{5}{6}}+C\)