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Question Number 135305 by JulioCesar last updated on 12/Mar/21

Answered by MJS_new last updated on 12/Mar/21

$$\int \frac{dx}{a^x+b}=$$$$[t=a^x+b\to dx=\frac{dt}{a^x\ln a}]$$$$=\frac{1}{\ln a}\int \frac{dt}{t(t-b)}=\frac{1}{b\ln a}\int (\frac{1}{t-b}-\frac{1}{t})dt=$$$$=\frac{1}{b\ln a}(\ln (t-b)-\ln t)=$$$$=\frac{1}{b\ln a}(\ln a^x-\ln (a^x+b))=$$$$=\frac{x}{b}-\frac{\ln (a^x+b)}{b\ln a}+C$$

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