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Section A.2 Rules for Integrals
Sums, differences, constant factors.
\begin{align*}
\int k f(x)\, dx \amp= k \int f(x)\, dx \\
\int f(x) \pm g(x)\, dx \amp= \int f(x)\, dx \pm \int g(x)\, dx
\end{align*}
Substitution.
\begin{align*}
\text{If }\int f(x)\, dx \amp= F(x) + C\\
\text{ then } \int f(g(x))g'(x)\, dx \amp= F(g(x)) + C\\
\text{That is, with } u = g(x),\\
\int f(u(x)) \frac{du}{dx} dx \amp= \int f(u) \, du
\end{align*}
Integration by Parts.
\begin{align*}
\int u(x) \frac{dv}{dx} dx \amp= u(x) v(x) - \int v(x) \frac{du}{dx} dx\\
\text{That is, }
\int u\, dv \amp= u v - \int v\, du
\end{align*}
