Evaluate the following integrals: e^x (1+e^ x )(2+e^x) dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{e}^\text{x}}{(1+\text{e}^{\text{x}})(2+\text{e}^\text{x})}\text{dx}$

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Answer

To evaluate the following integral follow tha steps:
Let $\text{e}^\text{x}=\text{t}$ therefore $\text{e}^\text{x}\text{dx = dt}$
Now
$\int\frac{\text{e}^{\text{x}}}{(1+\text{e}^\text{x})(2+\text{e}^\text{x})}\text{dx}=\int\frac{\text{dt}}{(1+\text{t})(2+\text{t})}$
$=\int\frac{\text{dt}}{(1+\text{t})}-\int\frac{\text{dt}}{(2+\text{t})}$
$=\ln|1+\text{t}|-\ln|2+\text{t}|+\text{C}$
$=\ln\bigg|\frac{1+\text{t}}{2+\text{t}}\bigg|+\text{C}$
$=\ln\bigg|\frac{1+\text{e}^{\text{x}}}{2+\text{e}^{x}}\bigg|+\text{C}$

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