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Question Number 177193 by peter frank last updated on 02/Oct/22

$$\int \frac{{\mathrm{x}}^{2021}}{{\mathrm{x}}^{2022}+{\mathrm{x}}^{4043}}\mathrm{dx}$$

Answered by cortano1 last updated on 02/Oct/22

$$\frac{1}{\mathrm{x}({\mathrm{x}}^{2021}+1)}=\frac{1}{\mathrm{x}}-\frac{{\mathrm{x}}^{2020}}{{\mathrm{x}}^{2021}+1}$$$$\mathrm{I}=\int \frac{\mathrm{dx}}{\mathrm{x}}-\int \frac{{\mathrm{x}}^{2020}}{{\mathrm{x}}^{2021}+1}$$$$\mathrm{I}=\ln \mid \mathrm{x}\mid -\frac{1}{2021}\int \frac{\mathrm{d}({\mathrm{x}}^{2021}+1)}{{\mathrm{x}}^{2021}+1}$$$$\mathrm{I}=\ln \mid \mathrm{x}\mid -\frac{\ln \mid {\mathrm{x}}^{2021}+1\mid }{2021}+\mathrm{c}$$$$\mathrm{I}=\ln \mid \frac{\mathrm{x}}{\sqrt[2021]{{\mathrm{x}}^{2021}+1}}\mid +\mathrm{c}$$

Commented by peter frank last updated on 02/Oct/22

$$\mathrm{thank}\mathrm{you}$$

Answered by som(math1967) last updated on 02/Oct/22

$$\int \frac{x^{2021}}{x^{2022}(1+x^{2021})}dx$$$$\int \frac{dx}{x(1+x^{2021})}$$$$\int \frac{x^{2020}dx}{x^{2021}(1+x^{2021})}$$$$let{x}^{2021}=t\Rightarrow x^{2020}dx=\frac{dt}{2021}$$$$\frac{1}{2021}\int \frac{dt}{t(t+1)}$$$$=\frac{1}{2021}\int \frac{dt}{t}-\frac{1}{2021}\int \frac{dt}{t+1}$$$$=\frac{1}{2021}[lnt-ln\mid t+1\mid ]+C$$$$=\frac{1}{2021}ln\mid \frac{x^{2021}}{x^{2021}+1}\mid +C$$$$$$

Commented by peter frank last updated on 02/Oct/22

$$\mathrm{thank}\mathrm{you}$$

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