# how to integrate (x^2 +2x +2)/ (-x^2 e^x)

## Expert Answers You need to use the linearity of integral such that:

int (x^2 +2x +2)/(-x^2 e^x) dx = int x^2/(-x^2)*e^(-x)dx - 2 int x/(x^2)e^(-x)dx - 2 int 1/(x^2)e^(-x)dx

int (x^2 +2x +2)/(-x^2 e^x) dx = - int e^(-x)dx - 2 int 1/x*e^(-x)dx - 2 int 1/(x^2)*e^(-x)dx

You need to solve the integral int 1/x*e^(-x)dx  using parts such that:

int udv = uv - int vdu

u = 1/x => du = -1/x^2 dx

dv = e^(-x)dx => v = -e^(-x)

int 1/x*e^(-x)dx = -(e^(-x))/x - int 1/x^2*e^(-x)dx

int (x^2 +2x +2)/(-x^2 e^x) dx = e^(-x) + 2(e^(-x))/x + 2int 1/x^2*e^(-x)dx- 2 int 1/(x^2)*e^(-x)dx

Reducing like terms yields:

int (x^2 +2x +2)/(-x^2 e^x) dx = e^(-x) + 2(e^(-x))/x + c

int (x^2 +2x +2)/(-x^2 e^x) dx = e^(-x)(1 + 2/x) + c

Hence, evaluating the given integral yields int (x^2 +2x +2)/(-x^2 e^x) dx = e^(-x)(1 + 2/x) + c.

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