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Special Functions Lecture 1

\[\begin{aligned} \int \, x + \sqrt{x} \; dx &= \frac{x^{2}}{2} + \frac{2x^{\frac{3}{2}}}{3}+c \\ \int \, \sin^{3}(2x)\cos(2x) \, dx &= \frac{1}{2}\,\frac{sin^{4}(2x)}{4} +c\end{aligned}\]

In general: \[\begin{aligned} \int \, f^{n}(x)f^{'}(x)\,dx &= \frac{f^{n+1}(x)}{n+1} + c \qquad where\ n\neq-1\\\\\end{aligned}\] In case \(n = -1\) \[\begin{aligned} \int\,\frac{f^{'}(x)}{f(x)}\,dx &= \ln|f(x)|+c\end{aligned}\]

\[\begin{aligned} f^{-1}(f(x)) = f(f^{-1}(x)) = x\end{aligned}\]

Important Example: \[\begin{aligned} e^{ln\,f(x)} = \ln\,e^{f(x)} = f(x)\end{aligned}\]

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