0-1-log-1-x-x-2-x-3-x-4-1-1-x-1-x-2-1-x-3-1-x-4-dx-

Question Number 77570 by TawaTawa last updated on 08/Jan/20

\int_{0}^{1} \log (\frac{1 + x + x^{2} + x^{3} + x^{4}}{\sqrt{1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}}}}) dx

Answered by MJS last updated on 08/Jan/20

= \underset{0}{\int^{1}} \ln (x^{2} \sqrt{x^{4} + x^{3} + x^{2} + x + 1}) d x =

= 2 \underset{0}{\int^{1}} \ln x d x + \frac{1}{2} \underset{0}{\int^{1}} \ln (x^{4} + x^{3} + x^{2} + x + 1) d x

2 \underset{0}{\int^{1}} \ln x d x = 2 {[x (\ln x - 1)]}_{0}^{1} =

[\lim_{x \to 0} x (\ln x - 1) = 0]

= - 2

\frac{1}{2} \underset{0}{\int^{1}} \ln (x^{4} + x^{3} + x^{2} + x + 1) d x =

[α_{j} = e^{i \frac{2 π j}{5}}; j = 1, 2, 3, 4]

= \frac{1}{2} \underset{0}{\int^{1}} \ln (\underset{j = 1}{\prod^{4}} (x - α_{j})) d x =

= \frac{1}{2} \underset{j = 1}{\sum^{4}} (\underset{0}{\int^{1}} \ln (x - α_{j}) d x) =

= \frac{1}{2} {[\underset{j = 1}{\sum^{4}} ((x - α_{j}) \ln (x - α_{j})) - 2 x]}_{0}^{1} = \dots

\dots = \frac{\sqrt{5}}{4} \ln \frac{1 + \sqrt{5}}{2} + \frac{5}{8} \ln 5 + \frac{\sqrt{25 + 10 \sqrt{5}}}{20} π - 2

\Rightarrow

answer is \frac{\sqrt{5}}{4} \ln \frac{1 + \sqrt{5}}{2} + \frac{5}{8} \ln 5 + \frac{\sqrt{25 + 10 \sqrt{5}}}{20} π - 4

Commented by TawaTawa last updated on 08/Jan/20

God bless you sir .

Commented by TawaTawa last updated on 12/Jan/20

Please sir . How did you get : e^{i \frac{2 π j}{5}}; j = 1, 2, 3, 4

Commented by MJS last updated on 12/Jan/20

(x^{4} + x^{3} + x^{2} + x + 1) (x - 1) = x^{5} - 1

\Rightarrow the roots are the 5^{th} roots of unity

= e^{i \frac{2 π j}{5}} with j = 0, 1, 2, 3, 4 but j = 0 \Leftrightarrow x - 1 = 0

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