# [SOLVED] Tech. of Integration

• Mar 22nd 2009, 09:47 AM
swensonm
[SOLVED] Tech. of Integration
xe^(2x)/ (2x+1)^2
• Mar 22nd 2009, 11:26 AM
Quote:

Originally Posted by swensonm
xe^(2x)/ (2x+1)^2

$\int{\frac{e^{2x}\times x}{(2x+1)^2}}$
-----------------------------
This is a simple question if you remember that
whenever you get an integral with e^x

Try this

$\int{e^{x} (f(x) +f'(x) )} = e^x f(x) + C$

Try proving it its fun (Happy)
-----------------------
Here let 2x =t

2dx =dt
dx =dt/2

(2x+1) = t+ 1

$
\int{\frac{e^{t}\times t dt }{4(t+1)^2}}
$

All you need to do now is See the form , try it before looking down

----------------------------------------
Here

$
\int{e^{t}~\frac{( t dt) }{4(t+1)^2}}
$

$
\int{e^{t}~\frac{(t+1-1) dt }{4(t+1)^2}}
$

$
\int{\frac{e^{t}}{4}(\frac{ t+1}{(t+1)^2}-\frac{ 1 }{(t+1)^2}})dt
$

Can you see something (Giggle)
-------------------------------------------------------------

$
\frac{d}{dt} \{\frac{ 1 dt }{(t+1)}\} = \frac{ -1}{(t+1)^2}
$

So ultimately
integration has become

$
\int{\frac{e^{t}}{4}(f'(t)+f(t))}
$

$e^{t} f(t)/4 = \frac{ e^{t} \{ \frac{ 1 }{ (t+1)} \} }{4}$

Put the value of t = 2x

$e^{2x} f(2x)/4 = \{ \frac{ e^{2x} } { 4(2x+1)} \}$
• Mar 22nd 2009, 07:25 PM
Jhevon
Quote:

$\int{\frac{e^{2x}\times x}{(2x+1)^2}}$
$\int{e^{x} (f(x) +f'(x) )} = e^x f(x) + C$