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Question Number 86518 by john santu last updated on 29/Mar/20

$$\int \frac{\mathrm{dx}}{{\mathrm{e}}^{2\mathrm{x}}-{5\mathrm{e}}^{\mathrm{x}}}$$

Commented by john santu last updated on 29/Mar/20

$$\Rightarrow \int \frac{\mathrm{dx}}{{\mathrm{e}}^{\mathrm{x}}({\mathrm{e}}^{\mathrm{x}}-5)}=\int \frac{{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}}{{\mathrm{e}}^{2\mathrm{x}}({\mathrm{e}}^{\mathrm{x}}-5)}$$$$\mathrm{let}\mathrm{t}={\mathrm{e}}^{\mathrm{x}}-5\Rightarrow \mathrm{dt}={\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$$$$\int \frac{\mathrm{dt}}{{(\mathrm{t}+5)}^2\mathrm{t}}=\int \frac{\mathrm{A}}{\mathrm{t}+5}\mathrm{dt}+\int \frac{\mathrm{B}}{{(\mathrm{t}+5)}^2}\mathrm{dt}$$$$+\int \frac{\mathrm{C}}{\mathrm{t}}\mathrm{dt}$$$$$$

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