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Question Number 159008 by amin96 last updated on 11/Nov/21

$$\int \frac{6^x}{4^x+9^x}dx=?$$

Answered by qaz last updated on 11/Nov/21

$$\int \frac{6^{\mathrm{x}}}{4^{\mathrm{x}}+9^{\mathrm{x}}}\mathrm{dx}$$$$=\int \frac{\mathrm{dx}}{{\left(\frac{2}{3}\right)}^{\mathrm{x}}+{\left(\frac{2}{3}\right)}^{-\mathrm{x}}}$$$$=\int \frac{{\left(\frac{2}{3}\right)}^{\mathrm{x}}}{{\left(\frac{2}{3}\right)}^{2\mathrm{x}}+1}\mathrm{dx}$$$$=\frac{1}{\ln \frac{2}{3}}\int \frac{\mathrm{d}\left({\left(\frac{2}{3}\right)}^{\mathrm{x}}\right)}{{\left(\frac{2}{3}\right)}^{2\mathrm{x}}+1}$$$$=\frac{1}{\ln \frac{2}{3}}\arctan {\left(\frac{2}{3}\right)}^{\mathrm{x}}+\mathrm{C}$$

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