Mathematics: How do you integrate [tex]rac{1}{e^x+2}[/tex]? using partial fractions (another method is ok). you!

This is simple:If k=3/5, then you substitute 3/5 for kExample:5(3/5)(-7)=? -175/3 or -58.33 repeating...

How do you integrate [tex]\frac{1}{e^x+2}[/tex]?

malakabdulah5792

14.01.2022

By substituting y=e^x+2 , or $x=\ln(y-2)$ so that $\mathrm dx=\frac{\mathrm dy}{y-2}$ , we have

$\displaystyle\int\frac{\mathrm dx}{e^x+2}=\int\frac{\mathrm dy}{y(y-2)}$

Split the integrand into partial fractions:

$\dfrac1{y(y-2)}=\dfrac12\left(\dfrac1{y-2}-\dfrac1y\right)$

Then integrating gives

$\displaystyle\int\frac{\mathrm dy}{y(y-2)}=\frac12(\ln|y-2|-\ln|y|)+C$

Back-substitute to get

$\displaystyle\int\frac{\mathrm dx}{e^x+2}=\frac12(\ln|e^x|-\ln|e^x+2|)+C=\frac{x-\ln(e^x+2)}2+C$

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