- xaxb -a-b- xbxc -b-c- xcxa -c-a - -

The correct option is A 1 (x^ax^b)^a+b(x^bx^c)^b+c(x^cx^a)^c+a ⇒x^a^2+abx^b^2+bcx^c^2+cax^ab+b^2x^bc+c^2x^ac+a^2 ⇒x^(a^2+b^2+c^2+ab+bc+ca)x^(a^2+ ...

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Special Integrals - 1

[ xaxb ]a+b×[...

Question

${(\begin{matrix} \frac{x^{a}}{x^{b}} \end{matrix})}^{a + b} \times {(\begin{matrix} \frac{x^{b}}{x^{c}} \end{matrix})}^{b + c} \times {(\begin{matrix} \frac{x^{c}}{x^{a}} \end{matrix})}^{c + a}$ = ____.

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Solution

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\frac{x^{a + b} \times x^{b + c} \times x^{c + a}}{(x^{a} \times x^{b} \times x^{c})}

{(\begin{matrix} \frac{x^{b}}{x^{c}} \end{matrix})}^{(b + c - a)} . {(\begin{matrix} \frac{x^{c}}{x^{a}} \end{matrix})}^{(c + a - b)} . x {(\begin{matrix} \frac{x^{a}}{x^{b}} \end{matrix})}^{(a + b - c)} = ?

\frac{1}{1 + x^{(b - a)} + x^{(c - a)}} + \frac{1}{1 + x^{(a - b)} + x^{(c - b)}} + \frac{1}{1 + x^{(b - c)} + x^{(a - c)}} = ?

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}} = 1

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} = 1

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{a - c} + x^{b - c}}

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