Evaluating the indefinite integral.

In summary, the given integral \int \frac{2x + 2}{x^2 + 2x + 5} \, dx can be simplified using the substitution u=x^2+2x+5 and the result can be expressed as \ln \left|x^2+2x+5\right|+\mathbb{C}. However, since the expression x^2+2x+5 is always non-negative for any real x, the use of absolute value signs is not necessary and the solution can be written simply as \ln(x^2+2x+5)+\mathbb{C}.
  • #1
shamieh
539
0
How does

\(\displaystyle \int \frac{2x + 2}{x^2 + 2x + 5} \, dx\) turn into \(\displaystyle \ln(x^2 + 2x + 5)\)?

How are they getting rid of the numerator are they just dividing by the reciprocal of 2x + 2?
 
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  • #2
The simplest thing to do is use the substitution:

\(\displaystyle u=x^2+2x+5\,\therefore\,du=(2x+2)\,dx\)

And now you have:

\(\displaystyle \int\frac{1}{u}\,du\)
 
  • #3
Hmm, this was the result of a partial fraction decomposition problem, I guess after solving three formulas a u substitution just seemed to simple (Tauri)(Dance)
 
  • #4
\(\displaystyle \int \frac{2x + 2}{x^2 + 2x + 5} \, dx\) turn into \(\displaystyle \ln(x^2 + 2x + 5)\)?

Solution:: Given $\displaystyle \int\frac{2x+2}{x^2+2x+5}dx$

Let $x^2+2x+5 = t$, Then $\left(2x+2\right)dx = dt$

So Integral convert into $\displaystyle \int\frac{1}{t}dt = \ln \left|t\right|+\mathbb{C}$

So $\displaystyle \int\frac{2x+2}{x^2+2x+5}dx = \ln \left|x^2+2x+5\right|+\mathbb{C}$
 
  • #5
My teacher has the solution has \(\displaystyle \ln(x^2 + 2x + 5)\) without the abs value signs...Do you think it's a typo on his part or is there any particular reasons I shouldn't put the abs value sign?
 
  • #6
shamieh said:
My teacher has the solution has \(\displaystyle \ln(x^2 + 2x + 5)\) without the abs value signs...Do you think it's a typo on his part or is there any particular reasons I shouldn't put the abs value sign?

Complete the square:

\(\displaystyle x^2+2x+5=(x+1)^2+2^2>0\) for all real $x$ so the absolute value signs are not necessary in this case. Your teacher is correct. :D
 
  • #7
With that being said, do you think I would get points taken off for just leaving it with the absolute value signs?
 
  • #8
shamieh said:
With that being said, do you think I would get points taken off for just leaving it with the absolute value signs?

Well, I can't speak for your professor, but technically either is correct. I don't think I would deduct points personally, however in my mind I would certainly favor the response that shows recognition of the fact that the expression is never negative for any real $x$.

However, let's put aside the notion of point deduction and look instead at the issue of mathematical simplicity and elegance. I think there is something to be said for striving to make our results as simple as possible while still covering all cases. I see putting absolute value signs around an expression that is always non-negative as a form of kludge. I think we should look at such things and remove anything that is unnecessary, so that our results are as lean as possible. (Wink)
 

FAQ: Evaluating the indefinite integral.

What is an indefinite integral?

An indefinite integral is a mathematical operation that is used to find the antiderivative of a function. It represents a family of functions that differ by a constant value.

How do you evaluate an indefinite integral?

The process of evaluating an indefinite integral involves finding the antiderivative of the given function. This can be done by using integration rules and techniques such as substitution, integration by parts, and partial fractions.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and gives a numerical value. An indefinite integral, on the other hand, does not have limits and represents a general family of functions.

Why is evaluating indefinite integrals important?

Evaluating indefinite integrals is important in mathematics because it allows us to find the original function from its derivative. It is also useful in solving real-world problems and in various fields such as physics, engineering, and economics.

What are some common techniques for evaluating indefinite integrals?

Some common techniques for evaluating indefinite integrals include substitution, integration by parts, trigonometric identities, and partial fractions. It is important to choose the appropriate technique based on the given function.

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