Find the derivative of

Question

Find the derivative of 

g'(x)=∫(u+4/u-5)du with upper bound 3x and lower bound 9x 

Answer 1

If the integrand is (u+4)/(u-5), you can rewrite it as 1+9/(u-5). The indefinite integral of this w.r.t. u is

 

u + 9ln|u-5|

 

Applying the limits of integration,

 

3x + 9ln|3x-5| - (9x + 9ln|9x-5|) = -6x + 9(ln|3x-5| - ln|9x-5|)

 

whose derivative (w.r.t x I'm assuming) is

 

-6 + 54(9x-10) / ( (3x-5)(9x-5) )