\hspace{-16}(1)\;\;\int\frac{25^x}{15^x-9^x}dx\\\\ (2)\;\;\int\frac{1}{1+x^6}dx\\\\ (3)\;\;\int\frac{\sqrt{1-x^2}-x}{x^3-x^2-x+1-\sqrt{1-x^2}+x\sqrt{1-x^2}}dx
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2 Answers
1(1)::\int \frac{\left ( \frac{25}{9} \right )^{x}}{\left ( \frac{15}{9} \right )^{x}-1} dx
\int \frac{\left ( \frac{5}{3} \right )^{2x}}{\left ( \frac{5}{3} \right )^{x}-1} dx
let\, \, \left ( \frac{5}{3} \right )^{x} =t \Rightarrow \left ( \frac{5}{3} \right )^{x} log\left ( \frac{5}{3} \right )dx =dt
integral\, \, is \, \, \frac{ 1}{log\left ( \frac{5}{3} \right )}\int \frac{t^{2}}{(t-1)t}dt
\frac{1}{log\left ( \frac{5}{3} \right )}\int \frac{tdt}{t-1}
which can be easily done
71For second, Partial Fractions method should work efficiently.
PS Note that I have posted answers from Wolfram only after solving it. It gives it in a neat form.
