Integrate {(secx tanx ) / (3secx +5)} dx.

\(int frac{3sec left(xright)tan left(xright)}{3sec left(xright)+5}dx\) \(mathrm{Take:the:constant:out}:quad int acdot fleft(xright)dx=acdot int fleft(xright)dx\) \(=3cdot int frac{sec left(xright)tan left(xright)}{3sec left(xright)+5}dx\)

Let u=sec(3x+5)

\(=3cdot int frac{1}{3u}du\) \(=3cdot frac{1}{3}cdot int frac{1}{u}du\) \(=3cdot frac{1}{3}ln left|uright|\) \(=3cdot frac{1}{3}ln left|3sec left(xright)+5right|\) \(=ln left|3sec left(xright)+5right|+C\)

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