Question

Proper solved answer

Don't spam otherwise report ​​

Answer 1

$\displaystyle \bold \red{\color{red}{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x}}}$

$\displaystyle{\bold \red{=\color{red}{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \cdot \tan{\left(x \right)}-\int{\tan{\left(x \right)} \cdot 2 e^{x} \cos{\left(x \right)} d x}\right)}}}$

${\displaystyle \bold \red{=\color{red}{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \int{2 e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \color{red}{\int{2 e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}} }}$

${ \displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \color{red}{\left(2 \int{e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}}}}$

${\displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(\sin{\left(x \right)} \cdot e^{x}-\int{e^{x} \cdot \cos{\left(x \right)} d x}\right)}}}$

$\displaystyle{\bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(e^{x} \sin{\left(x \right)} - \int{e^{x} \cos{\left(x \right)} d x}\right)}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\int{e^{x} \cos{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\left(\cos{\left(x \right)} \cdot e^{x}-\int{e^{x} \cdot \left(- \sin{\left(x \right)}\right) d x}\right)}}}$

${ \displaystyle{ \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\left(e^{x} \cos{\left(x \right)} - \int{\left(- e^{x} \sin{\left(x \right)}\right)d x}\right)}}}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} - 2 \color{red}{\int{\left(- e^{x} \sin{\left(x \right)}\right)d x}}}}$

${ \displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} - 2 \color{red}{\left(- \int{e^{x} \sin{\left(x \right)} d x}\right)}}}$

${ \displaystyle{ \bold \red{ \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle{\bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} + 2 \int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{\int{e^{x} \sin{\left(x \right)} d x}}} \\ \bold \red{= \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${\bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2}\right)}}}$

${\displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = - \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}}$

${ \displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = \frac{e^{x}}{\cos{\left(x \right)}}}}$

${\displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = \frac{e^{x}}{\cos{\left(x \right)}}+C}}$

Answer 2

Answer:

$\displaystyle \bold \red{\color{red}{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x}}}$

$\displaystyle{\bold \red{=\color{red}{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \cdot \tan{\left(x \right)}-\int{\tan{\left(x \right)} \cdot 2 e^{x} \cos{\left(x \right)} d x}\right)}}}$

${\displaystyle \bold \red{=\color{red}{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \int{2 e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \color{red}{\int{2 e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}} }}$

${ \displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - \color{red}{\left(2 \int{e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \cos{\left(x \right)} \tan{\left(x \right)} d x}}}}$

${\displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(\sin{\left(x \right)} \cdot e^{x}-\int{e^{x} \cdot \cos{\left(x \right)} d x}\right)}}}$

$\displaystyle{\bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(e^{x} \sin{\left(x \right)} - \int{e^{x} \cos{\left(x \right)} d x}\right)}}}$

${ \displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\int{e^{x} \cos{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\left(\cos{\left(x \right)} \cdot e^{x}-\int{e^{x} \cdot \left(- \sin{\left(x \right)}\right) d x}\right)}}}$

${ \displaystyle{ \bold \red{=\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 \color{red}{\left(e^{x} \cos{\left(x \right)} - \int{\left(- e^{x} \sin{\left(x \right)}\right)d x}\right)}}}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} - 2 \color{red}{\int{\left(- e^{x} \sin{\left(x \right)}\right)d x}}}}$

${ \displaystyle \bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} - 2 \color{red}{\left(- \int{e^{x} \sin{\left(x \right)} d x}\right)}}}$

${ \displaystyle{ \bold \red{ \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle{\bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 e^{x} \sin{\left(x \right)} + 2 e^{x} \cos{\left(x \right)} + 2 \int{e^{x} \sin{\left(x \right)} d x}}}}$

${ \displaystyle \bold \red{\int{e^{x} \sin{\left(x \right)} d x}}} \\ \bold \red{= \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2}}$

${\displaystyle \bold \red{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\int{e^{x} \sin{\left(x \right)} d x}}}}$

${\bold \red{= \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)} - 2 \color{red}{\left(\frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2}\right)}}}$

${\displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = - \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}}$

${ \displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = \frac{e^{x}}{\cos{\left(x \right)}}}}$

${\displaystyle \bold \red{\int{\left(\tan{\left(x \right)} + 1\right) e^{x} \sec{\left(x \right)} d x} = \frac{e^{x}}{\cos{\left(x \right)}}+C}}$