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1.10: Substitution

Last updated

Nov 5, 2021
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- 1.9: Indefinite Integrals of Elementary Functions
- 1.11: Integration by Parts

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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

$\begin{aligned}\int f'(g(x))g'(x)dx&=\int f'(y)dy \\ &=f(y)+c \\ &=f(g(x))+c.\end{aligned}$

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