change of variable – Solved Problems

Find $\displaystyle\int \! \frac{e^x}{1+e^x}\, dx$ .

Solution

Let $u=1+e^x$ , therefore $du=e^x dx \to dx=\frac{1}{u-1}du$ . Hence

$\displaystyle\int \! \frac{e^x}{1+e^x}\, dx = \displaystyle\int \! \frac{u-1}{u} \frac{1}{u-1}\,du = \displaystyle\int \! \frac{1}{u} \,du=\ln (u)=\ln(1+e^x)$

Tag: change of variable

Problem 2-8: Integration