Question

Integrate 1/(2x+3)

I know the answer but i don't see how to get it?

Is it a double U substitution or is it some identify I'm suppose to know
Please explain step by step
Thanks

Answer 1

use standard form integral of 1/ x = ln x

x is replaced by a linear function 2x + 3

the rule is to multiply by 1/coefficient of x in this case 1/2 and multiply this by ln ( 2x + 3)

so integral of 1/ 2x + 3 = 1/2* ln (2x + 3) + C

Answer 2

You are correct, how did u know to use 1/x though?
I would of never seen that

Answer 3

1 / (2x+3) is 1/x with the x replaced by (2x+3)
functions of the type 1/(ax +b) can all be integrated this way

you need to have this knowledge of standard form

Answer 4

notice how the integration "formula" is written as
\(\Large \int \frac{1}{u}du=lnu + C \) where u is a function of x ?

Answer 5

Thanks

Answer 6

it's knowing the above I mentioned, AND doing a substitution.....

Answer 7

let u=2x+3
so du=2dx ---> \(\frac {1}{2} \)du = dx

\(\Large \int \frac{1}{2x+3}dx = \int \frac{1}{u} \cdot (\frac{1}{2}du)=\frac{1}{2} \int \frac{1}{u}du \)