Thread: why doesn't the denominator power get integrated with result ending with a ln ?

1. why doesn't the denominator power get integrated with result ending with a ln ?

These two particular integrals are doing my head in ....

$$\int_{}^{} (4x+4)/(2x^2+4x+5) dx$$

which worked out to be
ln(2x^2 + 4x + 5) + c

and

$$\int_{}^{} (x+1)/(2x^2+4x+5) dx$$

which worked out to be

1/ 4ln(2x^2 + 4x + 5) + c

these where the answers in the worksheet answers ..I understand that when there's a fraction we need to us the natural log ...and then put the top x value over the bottom x value out the front of the natural log in the answer ,but the powers in the denominators have got me totally confused ....why don't we integrate that as well...?

2. Re: why doesn't the denominator power get integrated with result ending with a ln ?

You should know $\dfrac{d}{dx} \ln(u) = \dfrac{u'}{u}$, where $u$ is a function of $x$.

the antiderivative of $\dfrac{u'}{u}$ is $\ln|u| + C$ ... that's it.

For $\displaystyle \int \dfrac{x+1}{2x^2+4x+5} \, dx$, note the derivative of the denominator is $4x+4$. To get the numerator in that form, multiply the integrand by $\dfrac{4}{4}$ as follows ...

$\displaystyle \dfrac{1}{4} \int \dfrac{4(x+1)}{2x^2+4x+5} \, dx = \dfrac{1}{4} \ln(2x^2+4x+5) + C = \ln\sqrt[4]{2x^2+4x+5} + C$

note that absolute value is not necessary because $2x^2+4x+5 > 0$ for all $x$.

3. Re: why doesn't the denominator power get integrated with result ending with a ln ?

I must admit I am a bit stumped by d/dxln(u) = u'/u is a function of x.
the antiderivative of u'/u is ln|u| + C ....
the way I think I did it was using the x value on the numerator and extracting the x value from the denominator with the integrated chain rule...and making a fraction of it....ie 4/4 first integral and 1/4 second one....they came out correct as the part in front of the ln....hmmm is that way ok as well or was it a total fluke I achieved that result...thanks

4. Re: why doesn't the denominator power get integrated with result ending with a ln ?

$u=2x^2+4x+5$