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Question Number 146361 by gsk2684 last updated on 13/Jul/21

$$find\int \frac{1}{x^n+1}dxforn\in N$$

Answered by mathmax by abdo last updated on 13/Jul/21

$${\mathrm{z}}^{\mathrm{n}}+1=0\Rightarrow {\mathrm{z}}^{\mathrm{n}}={\mathrm{e}}^{(2\mathrm{k}+1)\mathrm{i}\pi }\Rightarrow {\mathrm{z}}_{\mathrm{k}}={\mathrm{e}}^{\frac{(2\mathrm{k}+1)\mathrm{i}\pi }{\mathrm{n}}}\mathrm{and}\mathrm{o}⩽\mathrm{k}⩽\mathrm{n}-1\Rightarrow$$$$\frac{1}{{\mathrm{z}}^{\mathrm{n}}+1}=\sum _{\mathrm{k}=0}^{\mathrm{n}-1}\frac{{\mathrm{a}}_{\mathrm{k}}}{\mathrm{z}-{\mathrm{z}}_{\mathrm{k}}}\mathrm{we}\mathrm{have}{\mathrm{a}}_{\mathrm{k}}=\frac{1}{{\mathrm{nz}}_{\mathrm{k}}^{\mathrm{n}-1}}=-\frac{{\mathrm{z}}_{\mathrm{k}}}{\mathrm{n}}\Rightarrow$$$$\frac{1}{{\mathrm{z}}^{\mathrm{n}}+1}=-\frac{1}{\mathrm{n}}\sum _{\mathrm{k}=0}^{\mathrm{n}-1}\frac{{\mathrm{z}}_{\mathrm{k}}}{\mathrm{z}-{\mathrm{z}}_{\mathrm{k}}}\Rightarrow \int \frac{\mathrm{dz}}{{\mathrm{z}}^{\mathrm{n}}+1}=-\frac{1}{\mathrm{n}}\sum _{\mathrm{k}=0}^{\mathrm{n}-1}{\mathrm{z}}_{\mathrm{k}}\log (\mathrm{z}-{\mathrm{z}}_{\mathrm{k}})+\mathrm{K}$$

Commented by gsk2684 last updated on 13/Jul/21

$$kindlyexplainsecondstep$$$$howcouldyoumakedecomposition$$

Commented by mathmax by abdo last updated on 13/Jul/21

$$\mathrm{if}\frac{\mathrm{p}\left(\mathrm{x}\right)}{\mathrm{q}\left(\mathrm{x}\right)}=\mathrm{\Sigma }\frac{{\mathrm{a}}_{\mathrm{i}}}{\mathrm{x}-{\mathrm{x}}_{\mathrm{i}}}\mathrm{with}\mathrm{degp}<\mathrm{degq}\mathrm{and}{\mathrm{x}}_{\mathrm{i}}\mathrm{simples}\mathrm{roots}$$$$\Rightarrow {\mathrm{a}}_{\mathrm{i}}=\frac{\mathrm{p}\left({\mathrm{x}}_{\mathrm{i}}\right)}{{\mathrm{q}}^{{}^{\prime }}\left({\mathrm{x}}_{\mathrm{i}}\right)}$$

Commented by gsk2684 last updated on 14/Jul/21

$$thankyou$$

Answered by Snail last updated on 13/Jul/21

$$\int \frac{x^{n-1}dx}{x^{n-1}(x^n+1)}$$$$=\frac{1}{n}\int \frac{dt}{1+\frac{1}{t}}where{x}^n=t\left(let\right)$$$$=\frac{1}{n}[\int dt-\int \frac{dt}{t+1}]$$$$=\frac{1}{n}[t-ln(t+1)+C]$$$$=\frac{1}{n}[x^n-ln(x^n+1)]+C$$

Commented by gsk2684 last updated on 14/Jul/21

$$howcouldyouwritesecondstep?please$$

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