$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$
$\int k f(x) dx = k \int f(x) dx$
-| $f(x)$ | $\int f(x) \cdot dx$ |
-| ------ | -------------------- |
-| $k$ (constant) | $kc + c$ |
-| $x^n (n \in J\\\{-1\})$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $f(x)$ | $\int f(x) \cdot dx$ |
+| ------------------------------- | ---------------------------- |
+| $k$ (constant) | $kc + c$ |
+| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $1 \over x$ | $\log_e x + c$ |
+| $e^kx$ | ${1 \over k} e^{kx} + c$ |
+| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
+| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
+| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
+| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
+| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
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