Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 162238 by amin96 last updated on 27/Dec/21

∫_0 ^1 (1/(x^7 +1))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{7}} +\mathrm{1}}\boldsymbol{\mathrm{dx}}=? \\ $$

Answered by Ar Brandon last updated on 27/Dec/21

z^7 +1=0⇒z_k =e^(((2k+1)/7)iπ) , k∈[0, 6], k∈Z  I=∫_0 ^1 (1/(x^7 +1))dx=∫_0 ^1 (dz/(Π_(k=0) ^6 (z−z_k )))=∫_0 ^1 Σ_(k=0) ^6 (a_k /(z−z_k ))dz  a_k =(1/(7z_k ^6 ))=−(z_k /7)  I=−(1/7)Σ_(k=0) ^6 ∫_0 ^1 (z_k /(z−z_k ))dz=−Σ_(k=0) ^6 [((z_k ln∣z−z_k ∣)/7)]_0 ^1

$${z}^{\mathrm{7}} +\mathrm{1}=\mathrm{0}\Rightarrow{z}_{{k}} ={e}^{\frac{\mathrm{2}{k}+\mathrm{1}}{\mathrm{7}}{i}\pi} ,\:{k}\in\left[\mathrm{0},\:\mathrm{6}\right],\:{k}\in\mathbb{Z} \\ $$$${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{7}} +\mathrm{1}}{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dz}}{\underset{{k}=\mathrm{0}} {\overset{\mathrm{6}} {\prod}}\left({z}−{z}_{{k}} \right)}=\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{k}=\mathrm{0}} {\overset{\mathrm{6}} {\sum}}\frac{{a}_{{k}} }{{z}−{z}_{{k}} }{dz} \\ $$$${a}_{{k}} =\frac{\mathrm{1}}{\mathrm{7}{z}_{{k}} ^{\mathrm{6}} }=−\frac{{z}_{{k}} }{\mathrm{7}} \\ $$$${I}=−\frac{\mathrm{1}}{\mathrm{7}}\underset{{k}=\mathrm{0}} {\overset{\mathrm{6}} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{z}_{{k}} }{{z}−{z}_{{k}} }{dz}=−\underset{{k}=\mathrm{0}} {\overset{\mathrm{6}} {\sum}}\left[\frac{{z}_{{k}} \mathrm{ln}\mid{z}−{z}_{{k}} \mid}{\mathrm{7}}\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com