Question

I'm not sure if this is arithmetic, properties of logs, or partial fraction decomposition, but whatever the case I'm stuck. The attached screenshot show the steps for the exercise, and it all makes sense to me EXCEPT the part where they turn 1/22 to 12/264. My question is why can't I leave it as 1/22, and what did I overlook for wanting to leave it as 1/22?

Answer 1

The reason it goes from \[\int\limits_{}^{}\frac{ 1 }{ 22 }\frac{ dx }{ 11+12x } + \int\limits_{}^{}\frac{ 1 }{ 22 }\frac{ dx }{ 11-12x }\] To \[\int\limits_{}^{}\frac{ 1 }{ 264 }\frac{ 12dx }{ 11+12x } + \int\limits_{}^{}\frac{ 1 }{ 264 }\frac{ 12dx }{ 11-12x }\] Is so that the top of the fraction is equual to the dervative of the bottom.

Answer 2

the integral of du/u is ln(u)

in this case u is 11+12x
u= 11+12x
take the derivative of both sides:
du = 12 dx

so you need a 12 dx up top. to compensate divide by 12

Answer 3

that makes sense to me @phi. thank you.