>885

(1)
　I = ∫ √(x^2 +1) dx
　　= x√(x^2 +1) - ∫{(x^2)/√(x^2 +1)} dx
　　= x√(x^2 +1) + ∫ {1/√(x^2 +1)} dx -I
　　= x√(x^2 +1) + log|x+√(x^2 +1)| -I,

　I = (1/2)x√(x^2 +1) + (1/2)log|x+√(x^2 +1)| +c.

(2)
　1/(1-x^2)^2 = (1/4){1/(1-x) +1/(1+x) +1/(1-x)^2 +1/(1+x)^2 },

　∫ {1/(1−x^2)^2} dx = (1/4){ log|(1+x)/(1-x)| +1/(1-x) -1/(1+x)} +c.

(3)
　1/{(x^3)(x^2 +1)} = (1-x^2)/(x^3) + x/(x^2 +1)= 1/(x^3) -1/x +x/(x^2 +1),

　∫ 1/{x^3(x^2＋1)} dx = -1/(2x^2) -log|x| +(1/2)log(x^2 +1) +c.