TITLE: Integration Substitution
QUESTION [5 upvotes]: I want to solve $\displaystyle \quad \int\frac{dz}{1+5z^2}$ 
and Wolfram says I need to use the substitution with inverse tan, taking $u=\sqrt{5}z$, giving me $$\frac{1}{\sqrt{5}}\int\frac{du}{1+u^2}.$$
Surely this is incorrect as the 1 in the demoninator would now be $\sqrt{5}$ too. What don't I understand?

REPLY [5 votes]: If $\;u = \sqrt{5}z\;$ then $\;du = \sqrt{5}\, dz \;\implies\; \color{red}{\bf{dz = \dfrac{1}{\sqrt5}} \,du}\;$ and
$$\int\frac{1}{1+5z^2}\,dz = \int \frac{1}{1 + (\sqrt{5}z)^2} \,\color{red}{\bf dz}=  \int\frac{1}{1+u^2} \cdot \color{red}{\bf{\frac{1}{\sqrt{5}}\,du}}\; = \;\dfrac{1}{\sqrt{5}} \int \dfrac{1}{1+u^2}\,du \;= \;\;?$$