Integral of (2x+1)/(x+2)^2 dx

• Jan 25th 2013, 03:09 PM
Riazy
Integral of (2x+1)/(x+2)^2 dx
I need to solve Integral of (2x+1)/(x+2)^2 dx
I would like to know how its throughly been done step by step, but before that I would atleast like to show you my progress even if it's a bit incorrent. The important thing is trying, but if someone could show me to complete steps I would be grateful

Progress:
1..The first thing I wanna see is the (x+2)^2 being split up into (x+2) (x+2)

so by having (2x+1)/ (x+2) (x+2)

2. Now looking for an expression of constants for I would suggest A/(x+2) + B/(x+2)
from here on I would kind of multiply everything up, to make it A(x+2) + B(x+2) / (x+2) (x+2)

3. This is where I get stuck, but im pretty sure its going to be an ln something + C

Could someone be kind and show me the steps?

// Best Regards
• Jan 25th 2013, 03:15 PM
MarkFL
Re: Integral of (2x+1)/(x+2)^2 dx
I think I would simply write:

$\displaystyle \frac{2x+1}{(x+2)^2}=\frac{(2x+4)-3}{(x+2)^2}=\frac{2(x+2)-3}{(x+2)^2}=\frac{2}{x+2}-\frac{3}{(x+2)^2}$
• Jan 25th 2013, 03:16 PM
Plato
Re: Integral of (2x+1)/(x+2)^2 dx
Quote:

Originally Posted by Riazy
I need to solve Integral of (2x+1)/(x+2)^2 dx
I would like to know how its throughly been done step by step, but before that I would atleast like to show you my progress even if it's a bit incorrent. The important thing is trying, but if someone could show me to complete steps I

See it done here . Click show steps tab.
• Jan 25th 2013, 05:32 PM
Prove It
Re: Integral of (2x+1)/(x+2)^2 dx
Quote:

Originally Posted by Riazy
I need to solve Integral of (2x+1)/(x+2)^2 dx
I would like to know how its throughly been done step by step, but before that I would atleast like to show you my progress even if it's a bit incorrent. The important thing is trying, but if someone could show me to complete steps I would be grateful

Progress:
1..The first thing I wanna see is the (x+2)^2 being split up into (x+2) (x+2)

so by having (2x+1)/ (x+2) (x+2)

2. Now looking for an expression of constants for I would suggest A/(x+2) + B/(x+2)
from here on I would kind of multiply everything up, to make it A(x+2) + B(x+2) / (x+2) (x+2)

3. This is where I get stuck, but im pretty sure its going to be an ln something + C

Could someone be kind and show me the steps?

// Best Regards

\displaystyle \displaystyle \begin{align*} \int{\frac{2x+1}{(x + 2)^2}\,dx} \end{align*}

Let \displaystyle \displaystyle \begin{align*} u = x + 2 \implies du = dx \end{align*} and note that

\displaystyle \displaystyle \begin{align*} u &= x + 2 \\ u - 2 &= x \\ 2u - 4 &= 2x \\ 2u - 3 &= 2x + 1 \end{align*}

So now we can rewrite the integral

\displaystyle \displaystyle \begin{align*} \int{\frac{2x + 1}{(x + 2)^2}\,dx} &= \int{\frac{2u - 3}{u^2}\,du} \\ &= \int{\frac{2}{u} - 3u^{-2}\,du} \\ &= 2\ln{|u|} + 3u^{-1} + C \\ &= 2\ln{|x + 2|} + \frac{3}{x + 2} + C \end{align*}