# Make a substitution to express the integrand as a rational function and then evaluate the integral. `int (e^(2x))/((e^(2x))+(3e^x)+2 )dx` ` ` choose an appropriate positive integer n in this...

Make a substitution to express the integrand as a rational function and then evaluate the integral.

`int (e^(2x))/((e^(2x))+(3e^x)+2 )dx`

` `

choose an appropriate positive integer n in this substitution u=`root(n)(x)`

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`int (e^(2x) dx) / (e^(2x) + 3e^x + 2)` has to be evaluated by proper substitution.

Let, `u = e^x`

`x = ln u `

and `1/udu = dx `

The integral then takes the form,

`int (u^2 * ((du)/u) )/( u^2 + 3u + 2)`

`= int (udu) / ((u+1)(u+2)) ` (factoring the denominator)

Decompose the rational expression into partial fractions:

Let `u/[(u+1)(u+2)] = A/(u+1) + B/(u+2)`

`rArr u = A(u+2) + B(u + 1) `

The two zeroing numbers are `u = -1` and `u=-2`

Put `u=-1` to obtain,

`-1 = A(1) +0`

`rArr A=-1 `

Again, put `u=-2` to obtain,

`-2 = 0 +B(-1)`

`rArr B=2`

So `u/[(u + 1)(u + 2)] = -1/(u + 1) + 2/(u + 2)`

And the integral now takes the form,

`int (-1/(u + 1) + 2/(u + 2))du`

`=-ln(u+1)+2ln(u+2)+C`

`=-ln(e^x+1)+2ln(e^x+2)+C` (Putting back the value of u)