∫ 1/(x2 + 4)2 dx
Substitution
x = 2·TAN(u)
dx = 2/COS(u)2 du
∫ 1/((2·TAN(u))2 + 4)2 dx
∫ 1/(4·SIN(u)2/COS(u)2 + 4)2 dx
∫ 1/(4·(SIN(u)2/COS(u)2 + 1))2 dx
∫ 1/(4·(SIN(u)2/COS(u)2 + COS(u)2/COS(u)2))2 dx
∫ 1/(4·((SIN(u)2 + COS(u)2)/COS(u)2))2 dx
∫ 1/(4·(1/COS(u)2))2 dx
∫ 1/(16·(1/COS(u)4)) dx
∫ COS(u)4/16 dx
∫ COS(u)4/16 · 2/COS(u)2 du
∫ COS(u)2/8 du
SIN(u)·COS(u)/16 + u/16
TAN(u)/(16 + 16·TAN(u)2) + u/16
Resubstitution
u = ARCTAN(x/2)
(x/2)/(16 + 16·(x/2)2) + ARCTAN(x/2)/16
(x/2)/(16 + 16·x2/4) + ARCTAN(x/2)/16
(x/2)/(4·x2 + 16) + ARCTAN(x/2)/16
x/(8·x2 + 32) + ARCTAN(x/2)/16