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6-x-4-x-9-x-dx-




Question Number 159008 by amin96 last updated on 11/Nov/21
∫(6^x /(4^x +9^x ))dx=?
$$\int\frac{\mathrm{6}^{{x}} }{\mathrm{4}^{{x}} +\mathrm{9}^{{x}} }{dx}=? \\ $$
Answered by qaz last updated on 11/Nov/21
∫(6^x /(4^x +9^x ))dx  =∫(dx/(((2/3))^x +((2/3))^(−x) ))  =∫((((2/3))^x )/(((2/3))^(2x) +1))dx  =(1/(ln(2/3)))∫((d(((2/3))^x ))/(((2/3))^(2x) +1))  =(1/(ln(2/3)))arctan ((2/3))^x +C
$$\int\frac{\mathrm{6}^{\mathrm{x}} }{\mathrm{4}^{\mathrm{x}} +\mathrm{9}^{\mathrm{x}} }\mathrm{dx} \\ $$$$=\int\frac{\mathrm{dx}}{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{x}} +\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{−\mathrm{x}} } \\ $$$$=\int\frac{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{x}} }{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2x}} +\mathrm{1}}\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{ln}\frac{\mathrm{2}}{\mathrm{3}}}\int\frac{\mathrm{d}\left(\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{x}} \right)}{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2x}} +\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{ln}\frac{\mathrm{2}}{\mathrm{3}}}\mathrm{arctan}\:\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{x}} +\mathrm{C} \\ $$

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