Find these indefinite integrals: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{1+x^4}dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x}{1+x^4}dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3}{1+x^4}dx$

Sketch the graph of the region whose area is given by the definite integral ${\displaystyle \underset{0}{\overset{9}{}}}\sqrt[\phantom{2}]{81-x^2}dx,$ then use a geometric formula to evaluate the area of the region.