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Question Number 124742 by TANMAY PANACEA last updated on 05/Dec/20

$$\int \frac{3^t+11}{6^t+11}dt\mathit{c}\mathit{o}\mathit{l}\mathit{l}\mathit{e}\mathit{c}\mathit{t}\mathit{e}\mathit{d}\mathit{p}\mathit{r}\mathit{o}\mathit{b}\mathit{l}\mathit{e}\mathit{m}$$

Commented by MJS_new last updated on 05/Dec/20

$$\mathrm{I}\mathrm{get}\mathrm{to}\mathrm{this}\mathrm{point}:$$$$\frac{3^t+11}{6^t+11}=\frac{1}{2^t}+\frac{11}{6^t+11}-\frac{11}{2^t(6^t+11)}=$$$$=\frac{1}{2^t}+\frac{6^t+11}{6^t+11}-\frac{6^t}{6^t+11}-\frac{11}{2^t(6^t+11)}=$$$$=\frac{1}{2^t}+1-\frac{6^t}{6^t+11}-\frac{11}{2^t(6^t+11)}$$$$\int \frac{dt}{2^t}+\int dt-\int \frac{6^t}{6^t+11}dt=-\frac{1}{2^t\ln 2}+t-\frac{\ln (6^t+11)}{\ln 6}$$$$\mathrm{but}\mathrm{no}\mathrm{idea}\mathrm{for}-11\int \frac{dt}{2^t(6^t+11)}$$

Commented by MJS_new last updated on 05/Dec/20

$$\mathrm{maybe}\mathrm{the}\mathrm{hypergeometric}\mathrm{function}\mathrm{is}$$$$\mathrm{possible}\mathrm{but}{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{not}\mathrm{good}\mathrm{at}\mathrm{using}\mathrm{it}...$$

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