# Calc1.55

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$\int _{{0}}^{{3}}{\frac {2x+1}{x^{2}+x+7}}\,dx\,$
To do a definite integral by substitution, the steps are all the same but you need to be careful. First, start out by finding a suitable $u\,$.
$u=x^{2}+x+7\,$
$du=(2x+1)dx\,$
Here is where you need to be careful. When $x=0\,$, we have $u=7\,$ so the limits of integration must be changed accordingly. Also, when $x=3\,$, we have $u=19\,$. Thus
$\int _{{0}}^{{3}}{\frac {2x+1}{x^{2}+x+7}}\,dx=\int _{{7}}^{{19}}{\frac {du}{u}}=\ln |u|{\bigg |}_{7}^{{19}}=\ln {\frac {19}{7}}\,$