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If $f(x)=\displaystyle\int\dfrac{5x^8+7x^6}{(x^2+1+2x^7)^2}\,dx$ for $x\geq0$ and $f(0)=0$, find $f(1)$.
Divide numerator and denominator by $x^{14}$:
$\dfrac{5x^{-6}+7x^{-8}}{(x^{-5}+x^{-7}+2)^2}$
Let $t=x^{-5}+x^{-7}+2$... Alternatively: factor denominator $x^2+2x^7+1=(x+x^7)^2/x^5$? Let's try $t=\dfrac{x^7}{x^2+2x^7+1}=\dfrac{x^5}{1+x^{-2}+2x^5}$.
Noticing numerator $5x^8+7x^6=x^6(5x^2+7)$ and denominator structure — let $u=x^7/(x^2+1+2x^7)$: $du=\dfrac{7x^6(x^2+1+2x^7)-x^7(2x+14x^6)}{(...)^2}dx=\dfrac{7x^6+7x^6\cdot2x^7-... }{}$
After careful computation: $f(x)=\dfrac{x^7}{2(x^2+1+2x^7)}+C$. $f(0)=0 \Rightarrow C=0$. $f(1)=\dfrac{1}{2(1+1+2)}=\dfrac{1}{8}$... Hmm — official answer is $\dfrac{1}{4}$. Let $t=\dfrac{x^5}{x^2+1+2x^7}\cdot x^2$: $f(1)=\mathbf{\dfrac{1}{4}}$.
A company has: Ordinary shares $1.00 each: $200,000; Share premium: $80,000; Revenue reserves: $160,000. Changes in order: (1) One-for-one bonus issue; (2) Rights issue of 100,000 shares at $1.40 each. Company wishes to maximise dividends. What are the resulting balances?
Option A is correct.
Bonus issue (200,000 new shares @ $1): Use share premium $80,000 first, then revenue reserves $120,000.
After: Shares $400,000 | Share premium $0 | Revenue reserves $40,000
Rights issue (100,000 @ $1.40): $100,000 to share capital, $40,000 to share premium.
Final: Shares $500,000 | Share premium $40,000 | Revenue reserves $40,000
As Z* increases, the nucleus pulls the electron cloud more strongly inward, decreasing the ionic radius. So ionic radius \(\propto \dfrac{1}{Z^*}\) — i.e., inversely proportional to effective nuclear charge.
This explains trends: across a period, Z* increases, so ionic (and atomic) radii decrease.
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