$$ \int \frac{\mathrm{e}^{\tan ^{-1} 2 x}}{1+4 x^2}= $$

$$ \int \mathrm{e}^x \frac{(x-1)}{(x+1)^3} \mathrm{~d} x= $$

$$ \int \sin ^5 x \mathrm{~d} x= $$

If $\int \frac{2 x+3}{(x-1)\left(x^2+1\right)} d x$

$$ =\log _e\left\{(x-1)^{\frac{5}{2}}\left(x^2+1\right)^2\right\}-\frac{1}{2} \tan ^{-1} x+\mathrm{A} $$

where A is an arbitrary constant, then the value of $a$ is