Integration of (3x+1) / (2x^2 - 2x +3 )

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In summary, to solve the given integral of (3x+1)/(2x^2-2x+3), you need to split it into two parts - one that will give the natural logarithm of the denominator, and one that can be turned into an inverse tangent by completing the square in the denominator. After solving, the final solution should no longer have an x in the numerator.
  • #1
teng125
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may i know how to solve this ques:
integra (3x+1) / (2x^2 - 2x +3 )
pls help...
 
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  • #2
This is a typical integral where you'll split it up in a part which will give the natural logarithm of the denominator: adjust the numerator so you get a part which is the denominator's derivative, you'll have a constant term too much and you split the integral there. The second one can then be turned into an inverse tangent by completing the square in the denominator.
 
  • #3
for the second part i got 3-x /(2x^2 - 2x +3)
may i know how to solve for inverse tangent??
 
  • #4
You're no longer supposed to have an x in the nominator, after you've split the ln-part.

The derivative of the denominator is 4x-2, it should look like this:

[tex]\int {\frac{{3x + 1}}{{2x^2 - 2x + 3}}dx} = \frac{3}{4}\int {\frac{{4x - 2 + 10/3}}{{2x^2 - 2x + 3}}dx} = \frac{3}{4}\int {\frac{{4x - 2}}{{2x^2 - 2x + 3}}dx} + \frac{{10}}{3}\frac{3}{4}\int {\frac{1}{{2x^2 - 2x + 3}}dx} [/tex]
 
  • #5
okok...got it...thanx
 
  • #6
Now we 'designed' the first integral so that it'll become the natural logarithm of the denominator (since we constructed the derivative in the nominator). The second integral now no longer has an x so you can complete the square in the denominator to form an arctan.
 

1. What is the process for integrating (3x+1) / (2x^2 - 2x +3)?

The process for integrating this expression involves using the method of partial fractions. This requires breaking the fraction into smaller, simpler fractions and then integrating each one individually.

2. Can the integral of (3x+1) / (2x^2 - 2x +3) be solved without using partial fractions?

Yes, it is possible to solve this integral without using partial fractions. An alternate method is to use the substitution technique, where you substitute a variable for the denominator and then solve the integral using the chain rule.

3. Is there a specific range of values for which the integral of (3x+1) / (2x^2 - 2x +3) is valid?

Yes, the integral is valid for all real values of x except for x=1 and x=-1. This is because the denominator becomes zero at these two values, making the integral undefined.

4. Is there a graphical representation of the integral of (3x+1) / (2x^2 - 2x +3)?

Yes, the integral can be represented graphically as the area under the curve of the given function. This area can be approximated using techniques such as the trapezoidal rule or Simpson's rule.

5. How can I check if my solution for the integral of (3x+1) / (2x^2 - 2x +3) is correct?

One way to check the solution is to differentiate it and see if it gives back the original function. Another method is to use online integration calculators or software to compare the result.

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