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sashaice [31]
1 year ago
15

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

(x) dx
Mathematics
1 answer:
Korolek [52]1 year ago
5 0

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{6x\sec(x) - 6 \ln|\sec(x) + \tan(x)| + C}

where the last integral follows from

\displaystyle \int \sec(x) \, dx = \int \frac{\sec(x)(\sec(x)+\tan(x))}{\sec(x)+\tan(x)} \, dx = \int \frac{d(\sec(x)+\tan(x))}{\sec(x)+\tan(x)}

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