Question

Evaluate: $\frac{x^p}{x^p+x^q}+\frac{1}{x^{p-q}+1}.$

Answer 1

Let $a=\frac{x^p}{x^p+x^q}+\frac{1}{x^{p-q}+1}.$

$=\frac{x^p}{x^p+x^q}+\frac{1}{\frac{x^p}{x^q}+1}$

$=\frac{x^p}{x^p+x^2}+\frac{x^q}{x^p+x^q}=\frac{x^p+x^q}{x^p+x^q}$

$a=1$

$\therefore \frac{x^p}{x^p+x^q}+\frac{1}{x^{p-q}+1}=1$