How do you integrate int (x+5)/(2x+3) using substitution?

1 Answer
Aug 23, 2016

=7/4ln(2x+3) + 1/2x + C

Explanation:

We can't immediately substitute into this integrand. First we have to get it into a more receptive form:

We do this with polynomial long division. It's a very simple thing to do on paper but the formatting is quite difficult on here.

int (x+5)/(2x+3)dx = int (7/(2(2x+3)) + 1/2)dx

=7/2int (dx)/(2x+3) + 1/2intdx

Now for the first integral set u = 2x+3 implies du = 2dx

implies dx = (du)/2

=7/4int(du)/(u) + 1/2intdx

=7/4ln(u) + 1/2x + C

=7/4ln(2x+3) + 1/2x + C