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List of antiderivatives

$$\int x^ndx=\frac{x^{n+1}}{n+1}+\Math{C}\;;\int f^n(x)f'(x)dx=\frac{f^{n+1}(x)}{n+1}+\Math{C}$$
$$\int \frac{1}{x}dx=\ln|x|+\Math{C}\;;\int \frac{f'(x)}{f(x)}dx=\ln|f(x)|+\Math{C}$$
$$\int \sin xdx=-\cos x+\Math{C}\;;\int \sin f(x)f'(x)dx=-\cos f(x)+\Math{C}$$
$$\int \frac{1}{\sin x}dx=\int \frac{1}{2\sin \frac{x}{2}\cos\frac{x}{2}}dx=\int \frac{1}{2\tan\frac{x}{2}\cos^2\frac{x}{2}}dx=\ln|\tan\frac{x}{2}|+\Math{C}$$
$$\int \cos xdx=\sin x +\Math{C}\;;\int \cos f(x)f'(x)dx=\sin f(x)+\Math{C}$$

Circumcircles of orthogonal four points

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