0.10: Substitution
More complicated functions can be integrated using the chain rule. Since
\[\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x),\nonumber \]
we have
\[\int f'(g(x))\cdot g'(x)dx=f(g(x))+c.\nonumber \]
This integration formula is usually implemented by letting \(y = g(x)\). Then one writes \(dy = g'(x)dx\) to obtain
\[\begin{aligned} \int f'(g(x))g'(x)dx&=\int f'(y)dy \\ &=f(y)+c \\ &=f(g(x))+c.\end{aligned} \nonumber \]