∫xe^(-x) dx
=-∫x de^(-x)
=-xe^(-x)+∫e^(-x) dx
=-xe^-x-e^(-x)+C
=-(x+1)e^(-x)+C
∫e^x/(1+e^2x) dx
=∫1/(1+e^2x) d(e^x)
=arctan(e^x)+C
∫1/(x²+2x+2) dx
=∫1/[(x+1)²+1] d(x+1)
=arctan(x+1)+C
∫1/[x√(2x²-1)] dx
=∫1/[2x²√(2x²-1)] d(x²)
=∫1/[2x²*2√(2x²-1)] d(2x²-1)
=∫1/(2x²) d√(2x²-1)
=∫1/{1+[√(2x²-1)]²} d√(2x²-1)
=arctan√(2x²-1)+C
∫arctanx/x² dx
=-∫arctanx d(1/x)
=-arctanx/x+∫1/x d(arctanx)
=-arctanx/x+∫1/[x(1+x²)] dx
=-arctanx/x+∫1/x dx-∫x/(1+x²) dx
=-arctanx/x+ln|x|-(1/2)∫1/(1+x²) d(1+x²)
=-arctanx/x+ln|x/(1+x²)|+C