سلسلة التكامل

$Q46)\int e^{x}(\frac{x+2}{x+4})^{2}dx$

$Q47)\int e^{arctanx}(\frac{1+x+x^{2}}{1+x^{2}})dx$

$Q46)\int e^{x}(\frac{x+2}{x+4})^{2}dx$

$Q47)\int e^{arctanx}(\frac{1+x+x^{2}}{1+x^{2}})dx$

\[Q46)\int e^{x}\left ( \frac{x+2}{x+4} \right )^{2}dx=\int e^{x}\left ( 1-\frac{2}{x+4} \right )^{2}dx\\=\int e^{x}\left ( 1-\frac{4}{x+4}+\frac{4}{\left ( x+4 \right )^{2}} \right )dx\\=\int \left ( e^{x}-4\times \frac{(x+4)e^{x}-e^{x}}{\left ( x+4 \right )^{2}} \right )dx\\=e^{x}-\frac{4e^{x}}{x+4}+C=\frac{xe^{x}}{x+4}+C\]

\[Q47)\int e^{arctan}\left ( \frac{1+x+x^{2}}{1+x^{2}} \right )dx=\\\int \left ( e^{arctanx}+\frac{xe^{arctanx}}{1+x^{2}} \right )dx=\\\int\left ( e^{arctanx}\left ({x}' \right ) +x{\left (e^{arctanx} \right )}'\right )dx=\\xe^{arctanx}+C\]