MHB Jamie's question at Yahoo Answers regarding an indefinite integral

AI Thread Summary
The discussion focuses on integrating the function (3x + 2)/(x^2 + 3x + 1) dx using different methods. One approach involves partial fraction decomposition, where the integrand is expressed in terms of its roots derived from the quadratic formula. Another method utilizes substitution, transforming the integral into a more manageable form. Both methods ultimately lead to the same result, showcasing the versatility of integration techniques. The final integral expressions highlight the importance of logarithmic and hyperbolic functions in solving such problems.
MarkFL
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Here is the question:

Integrate (3x + 2)/(x^2+3x+1) dx?

Please tell me the name of the way you did it, for ex. Start with integration by parts, or substitution.

I got to the Integral of 1-(3u+2)/(u^2+3u+1)du where u = tanx

I have posted a link there to this thread so the OP can view my work.
 
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Hello Jamie,

We are given to evaluate:

$$I=\int\frac{3x+2}{x^2+3x+1}\,dx$$

I would look at the partial fraction decomposition of the integrand. Application of the quadratic formula gives us the roots of the denominator as:

$$x=\frac{-3\pm\sqrt{5}}{2}$$

Hence, we may state:

$$x^2+3x+1=\frac{1}{4}\left(2x+3-\sqrt{5} \right)\left(2x+3+\sqrt{5} \right)$$

And so the integrand may be expressed as:

$$\frac{4(3x+2)}{\left(2x+3-\sqrt{5} \right)\left(2x+3+\sqrt{5} \right)}$$

Thus, we may assume the partial fraction of this integrand will take the form:

$$\frac{4(3x+2)}{\left(2x+3-\sqrt{5} \right)\left(2x+3+\sqrt{5} \right)}=\frac{A}{2x+3-\sqrt{5}}+\frac{B}{2x+3+\sqrt{5}}$$

Using the Heaviside cover-up method, we find:

$$A=\frac{4\left(3\left(\dfrac{-3+\sqrt{5}}{2} \right)+2 \right)}{2\left(\dfrac{-3+\sqrt{5}}{2} \right)+3+\sqrt{5}}=3-\sqrt{5}$$

$$B=\frac{4\left(3\left(\dfrac{-3-\sqrt{5}}{2} \right)+2 \right)}{2\left(\dfrac{-3-\sqrt{5}}{2} \right)+3-\sqrt{5}}=3+\sqrt{5}$$

And so we find:

$$\frac{4(3x+2)}{\left(2x+3-\sqrt{5} \right)\left(2x+3+\sqrt{5} \right)}=\frac{3-\sqrt{5}}{2x+3-\sqrt{5}}+\frac{3+\sqrt{5}}{2x+3+\sqrt{5}}$$

And we may now state:

$$I=\frac{3-\sqrt{5}}{2}\int\frac{2}{2x+3-\sqrt{5}}\,dx+\frac{3+\sqrt{5}}{2}\int\frac{2}{2x+3+\sqrt{5}}\,dx$$

Using the integration rule:

$$\int\frac{du}{u+a}\,du=\ln|u+a|+C$$

we find:

$$I=\frac{3-\sqrt{5}}{2}\ln|2x+3-\sqrt{5}|+\frac{3+\sqrt{5}}{2}\ln|2x+3+\sqrt{5}|+C$$

$$I=\frac{1}{2}\left((3-\sqrt{5})\ln|2x+3-\sqrt{5}|+(3+\sqrt{5})\ln|2x+3+\sqrt{5}| \right)+C$$

And in conclusion, we may now state:

$$\int\frac{3x+2}{x^2+3x+1}\,dx=\frac{1}{2}\left((3-\sqrt{5})\ln|2x+3-\sqrt{5}|+(3+\sqrt{5})\ln|2x+3+\sqrt{5}| \right)+C$$
 
MarkFL said:
Here is the question:
I have posted a link there to this thread so the OP can view my work.

When I look at integration problems, the first thing I always look for are simple substitutions...

$\displaystyle \begin{align*} \int{\frac{3x + 2}{x^2 + 3x + 1}\,dx} &= 3\int{\frac{x + \frac{2}{3}}{x^2 + 3x + 1} \, dx} \\ &= \frac{3}{2} \int{ \frac{2x + \frac{4}{3}}{x^2 + 3x + 1} \, dx } \\ &= \frac{3}{2} \int{\frac{2x + 3 - \frac{5}{3}}{x^2 + 3x + 1}\,dx} \\ &= \frac{3}{2} \int{ \frac{2x + 3}{x^2 + 3x + 1}\,dx } - \frac{3}{2}\int{\frac{\frac{5}{3}}{x^2 + 3x + 1} \, dx} \\ &= \frac{3}{2} \int{ \frac{2x + 3}{x^2 + 3x + 1}\, dx} - \frac{5}{2} \int{ \frac{1}{x^2 + 3x + \left( \frac{3}{2} \right) ^2 - \left( \frac{3}{2} \right) ^2 + 1 } \, dx} \\ &= \frac{3}{2} \int{ \frac{2x + 3}{x^2 + 3x + 1} \, dx} - \frac{5}{2} \int{ \frac{1}{ \left( x + \frac{3}{2} \right) ^2 - \frac{5}{4} } \, dx} \end{align*}$

Now making the substitutions $\displaystyle \begin{align*} u = x^2 + 3x + 1 \implies du = \left( 2x + 3 \right) \, dx \end{align*}$ and $\displaystyle \begin{align*} x + \frac{3}{2} = \frac{\sqrt{5}}{2}\cosh{(t)} \implies dx = \frac{\sqrt{5}}{2}\sinh{(t)}\,dt \end{align*}$ and we get

$\displaystyle \begin{align*} \frac{3}{2} \int{ \frac{2x + 3}{x^2 + 3x + 1} \, dx} - \frac{5}{2} \int{ \frac{1}{ \left( x + \frac{3}{2} \right) ^2 - \frac{5}{4} } \, dx} &= \frac{3}{2} \int{ \frac{1}{u}\,du } - \frac{5}{4} \int{ \frac{1}{\left[ \frac{\sqrt{5}}{2} \cosh{(t)} \right] ^2 - \frac{5}{4}} \, \frac{\sqrt{5}}{2}\sinh{(t)}\,dt } \\ &= \frac{3}{2} \ln{|u|} - \frac{5\sqrt{5}}{8} \int{ \frac{\sinh{(t)}}{\frac{5}{4} \left[ \cosh^2{(t)} - 1 \right]}\,dt } \\ &= \frac{3}{2} \ln{ \left| x^2 + 3x + 1 \right| } - \frac{\sqrt{5}}{2}\int{ \frac{\sinh{(t)}}{\sinh{(t)}}\,dt } \\ &= \frac{3}{2}\ln{ \left| x^2 + 3x + 1 \right| } - \frac{\sqrt{5}}{2} \int{ 1 \, dt} \\ &= \frac{3}{2} \ln{ \left| x^2 + 3x + 1 \right| } - \frac{\sqrt{5}}{2} t + C \\ &= \frac{3}{2} \ln{ \left| x^2 + 3x + 1 \right| } - \frac{\sqrt{5}}{2} \,\textrm{arcosh}\, { \left[ \frac{\sqrt{5} \, \left( 2x + 3 \right) }{5} \right] } + C \end{align*}$
 
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