# 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 ?

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