Exponential and logarithmic substitutions

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

A rational function of eˣ becomes rational in t=eˣ with t>0. When dx/x appears with logarithms, t=ln x often simplifies it on x>0.

Step-by-step example

1. Use t = eˣ.

   $$I=\int\frac{e^x}{1+e^x}\,dx,\quad dt=e^x\,dx$$

2. Integrate the rational fraction.

   $$I=\int\frac{dt}{1+t}=\ln(1+t)$$

3. Return to x; 1+eˣ is positive.

   $$I=\ln(1+e^x)+C$$