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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\,.



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}}\,

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