∫ 1 a x + b d x = 1 a l n | a x + b | + K = 1 a l n A | a x + b | Can anyone explain how the equation is true, and what the motivation for doing this is?

Question

∫      1          a      x      +      b        d  x  =      1    a    l  n      |    a  x  +  b      |    +  K  =      1    a    l  n  A      |    a  x  +  b      |  Can anyone explain how the equation is true, and what the motivation for doing this is?

Phillip Fletcher · Accepted Answer

If   K  =      1    a        log    e    ⁡  A, i.e. letting   A  =      e          a      K      , then you should be able to combine the logs.

Alexander Lewis · Accepted Answer

It follows from the fact that   ln  ⁡  u  v  =  ln  ⁡  u  +  ln  ⁡  v. In this case set   A  =      e    K   and take u=A and   v  =      |    a  x  +  b      |  : then   ln  ⁡  A      |    a  x  +  b      |    =  ln  ⁡  A  +  ln  ⁡      |    a  x  +  b      |    =  ln  ⁡      e    K    +  ln  ⁡      |    a  x  +  b      |    =  K  +  ln  ⁡      |    a  x  +  b      |

∫1ax+bdx=1aln|ax+b|+K=1alnA|ax+b| Can anyone explain how the equation is true, and what the motivation for doing this is?

Answered question

Answer & Explanation

New Questions in Calculus 1