Mathematics

Evaluate
$$ \int {  { e }^{ x }(\tan { x } +1)secx\quad dx}$$


SOLUTION
$$\begin{array}{c}I = \int {{e^x}\left( {\tan x + 1} \right)\sec xdx} \\ = \int {{e^x}\tan x\sec x} dx + \int {{e^x}\sec x} dx\end{array}$$

According to the integration by parts,
$$\begin{array}{l}\int {uvdx}  = u\int {vdx}  - \int {\left( {\frac{{du}}{{dx}} \times \int {vdx} } \right)} dx\\Here,\\u = \sec x\\v = {e^x}\\So,\\\int {\sec x{e^x}dx}  = \sec x\int {{e^x}dx}  - \int {\left( {\frac{{d\sec x}}{{dx}} \times \int {{e^x}dx} } \right)} dx\\ = {e^x}\sec x - \int {\left( {\sec x\tan x{e^x}} \right)} dx\\So,\\I = \int {\left( {\sec x\tan x{e^x}} \right)} dx + {e^x}\sec x - \int {\left( {\sec x\tan x{e^x}} \right)} dx\\ = {e^x}\sec x\end{array}$$
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Subjective Medium Published on 17th 09, 2020
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