integration in calculus

Question

Find or evaluate the integral by completing the square.

 

∫ x/(x⁴+10x²+26) dx

Answer 1

x⁴+10x²+26 = x⁴+10x²+25+1

 

                  = (x²+5)²+1

 

So, the given integral is equivalent to ∫[x/((x²+5)²+1)]dx

 

                                                          Let u = x²+5  so, du = 2xdx

 

                                                                                xdx = (1/2)du

 

∫[x/((x²+5)²+1)]dx = (1/2)∫[1/(1+u²)]du

 

                            = (1/2)Arctanu + C

 

                            = (1/2)Arctan(x²+5) + C