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evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) 6x sec(x) tan(x) dx

User AkshayBandivadekar
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1 Answer

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Integrate by parts.


\displaystyle \int 6x \sec(x) \tan(x) \, dx = uv - \int v\,du

with


u = 6x \implies du = 6\,dx


dv = \sec(x)\tan(x)\,dx \implies v = \sec(x)

so that


\displaystyle \int 6x \sec(x) \tan(x) \, dx = 6x\sec(x) - 6 \int \sec(x) \, dx \\\\ ~~~~~~~~ = \boxed\sec(x) + \tan(x)

where the last integral follows from


\displaystyle \int \sec(x) \, dx = \int (\sec(x)(\sec(x)+\tan(x)))/(\sec(x)+\tan(x)) \, dx = \int (d(\sec(x)+\tan(x)))/(\sec(x)+\tan(x))

User Dangowans
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