nice-calculus-suppose-that-p-0-ln-1-x-p-x-2-find-the-value-of-1-0-p-1-p-dp-

Question Number 136626 by mnjuly1970 last updated on 24/Mar/21

\dots \dots \dots . n i c e c a l c u l u s \dots \dots \dots

s u p p o s e t h a t :::

φ (p) = \int_{0}^{\infty} \frac{l n (1 + x)}{{(p + x)}^{2}} \dots ✓

f i n d t h e v a l u e o f ::

{\int_{0}}^{1} \frac{φ (p)}{1 + p} d p = ? \dots

Answered by mindispower last updated on 24/Mar/21

φ (p) = \int_{0}^{\infty} \frac{d x}{(p + x) (1 + x)} = \frac{1}{p - 1} {[l n (\frac{1 + x}{p + x})]}_{0}^{\infty}

= \frac{l n (p)}{p - 1}

\int_{0}^{1} \frac{- l n (p)}{(1 - p^{2})} d p = - \int_{0}^{1} \sum_{n ⩾ 0} p^{2 n} l n (p) d p

= Σ \frac{1}{{(2 n + 1)}^{2}} = \frac{3}{4} ζ (2) = \frac{π^{2}}{8}

Commented by mnjuly1970 last updated on 24/Mar/21

t a s h a k o r . g r a t e f u l . . s i r p o w e r . .