Q-Calculus-If-n-0-1-x-2n-1-x-2-dx-then-find-the-value-of-

Question Number 144787 by mnjuly1970 last updated on 29/Jun/21

You can't use 'macro parameter character #' in math mode

If : \boldsymbol ϕ (n) : = \int_{0}^{1} \frac{x^{2 n}}{1 + x^{2}} d x

then find the value of ::

S := \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} \boldsymbol ϕ (n)}{n} = ?

m . n . july . 1970

Answered by qaz last updated on 29/Jun/21

S = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n} ϕ (n)

= \int_{0}^{1} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n} \cdot \frac{x^{2 n}}{1 + x^{2}} dx

= \int_{0}^{1} \frac{\ln (1 + x^{2})}{1 + x^{2}} dx

= \int_{0}^{\infty} \frac{\ln (1 + x^{2})}{1 + x^{2}} dx - \int_{1}^{\infty} \frac{\ln (1 + x^{2})}{1 + x^{2}} dx

= - 2 \int_{0}^{π / 2} lncos udu - \int_{0}^{1} \frac{\ln (1 + x^{2}) - 2 lnx}{1 + x^{2}} dx

= π \ln 2 - S - 2 G

\Rightarrow S = \frac{π}{2} \ln 2 - G

Commented by mnjuly1970 last updated on 29/Jun/21

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