0\; \; \Rightarrow \; \; |x|=x\; \Big ]=\int \frac{dx}{x\cdot \sqrt{1-log^2_{a}x}}=\\\\=\Big [\; t=log_{a}x\; ,\; dt=\frac{dx}{x\cdot lna}\; \Big ]=lna\cdot \int \frac{dt}{\sqrt{1-t^2}}=\\\\=lna\cdot arcsint+C=lna\cdot arcsin(log_{a}x)+C" alt="\int \frac{dx}{\sqrt{x^2-x^2\, log^2_{a}x}}=\int \frac{dx}{\sqrt{x^2\cdot (1-log^2_{a}x)}}=\int \frac{dx}{|x|\cdot \sqrt{1-log^2_{a}x}}=\int \frac{dx}{\pm x\cdot \sqrt{1-log^2_{a}x}}=\\\\=\Big [\; log_{a}x\; \; \to \; \; x>0\; \; \Rightarrow \; \; |x|=x\; \Big ]=\int \frac{dx}{x\cdot \sqrt{1-log^2_{a}x}}=\\\\=\Big [\; t=log_{a}x\; ,\; dt=\frac{dx}{x\cdot lna}\; \Big ]=lna\cdot \int \frac{dt}{\sqrt{1-t^2}}=\\\\=lna\cdot arcsint+C=lna\cdot arcsin(log_{a}x)+C" align="absmiddle" class="latex-formula">