Paul Gallegos

2023-03-13

What is the integral of $sec\left(x\right)$?

svikaoeol

Find integration of $\mathrm{sec}\left(x\right)$.
To rephrase the integral:
$\int \mathrm{sec}xdx=\int \mathrm{sec}x\left(\frac{\mathrm{sec}x+\mathrm{tan}x}{\mathrm{sec}x+\mathrm{tan}x}\right)dx\int \mathrm{sec}xdx=\int \frac{{\mathrm{sec}}^{2}x+\mathrm{sec}x\mathrm{tan}x}{\mathrm{sec}x+\mathrm{tan}x}dx...\left(1\right)$
$\mathrm{Take},\mathrm{sec}x+\mathrm{tan}x=t⇒\left(\mathrm{sec}x\mathrm{tan}x+{\mathrm{sec}}^{2}x\right)dx=dt$.
Add the aforementioned values to the equation$\left(1\right)$ to find :
$\int \mathrm{sec}xdx=\int \frac{dt}{t}\int \mathrm{sec}xdx=\mathrm{ln}|t|+C\left(\because \int \frac{dt}{t}=\mathrm{ln}|t|\right)\int secxdx=\mathrm{ln}|\left(secx+\mathrm{tan}x\right)|+C\left(\mathrm{Put}t=\mathrm{sec}x+\mathrm{tan}x\right)$
Thus, the required integral is $\int secxdx=\mathrm{ln}|\left(secx+\mathrm{tan}x\right)|+C$ where $C$ is a constant of integration .

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