Need help integrating (4x - 3) / (x^2 +1) for my Maths Baccalaureate exam!

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In summary: I hadn't heard the term before so I googled it. You can check your abilities to integrate with the Wolfram integrator!
  • #1
jasper10
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Homework Statement



Can someone help me integrate (4x - 3) / (x^2 +1) ?

The Attempt at a Solution



I'm doing my maths baccalaureate in 2 days and came across this question today!

I don't know where to start i.e. which method to use:

Integration by parts definitely does not work, nor does by substitution!

Thank you!
 
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  • #2
Hi jasper10! :smile:

Trig subsitution should do it.

(alternatively, split it into the 4x part and the 3 part … you should be able to integrate those anyway :wink:)
 
  • #3
easy to do if you use integration by parts since the derivative of the numerator is a constant :D
 
  • #4
I think that tiny-tim's advice of splitting the integral into two parts is much simpler than integration by parts - hence, less opportunity for getting things fouled up.
 
  • #5
Unless I'm sorely mistaken integration by parts is one of the key things the maths part of an IB is supposed to teach you. I think my hint was sufficient and that if appropriate care is taken nothing should get 'fouled-up'.
 
  • #6
Ok so i get:

∫4x / (x^2 + 1) dx - ∫3/ (x^2 + 1) dx

= 2ln(x^2 + 1) - ?

How would you integrate: 3 / (x^2 + 1)?

the annoying part is the "x^2"

thanks!

ps: what do you mean by "trig" substitution?

(I have a feeling this type of question isn't on the syllabus, but i found it on a past back from 2002 - strange)
 
  • #7
DJsTeLF said:
Unless I'm sorely mistaken integration by parts is one of the key things the maths part of an IB is supposed to teach you. I think my hint was sufficient and that if appropriate care is taken nothing should get 'fouled-up'.

Integration by substitution surely cannot work!

it's a vicious circle and keeps getting more complicated whilst integrating
 
  • #8
I'm not disputing the importance of integration by parts. What I'm saying is that it's always a good idea to use the simplest technique that seems likely to work. For this problem, integration by parts ain't it.
 
  • #9
Mark44 said:
For this problem, integration by parts ain't it.

Yeah just had another look at it and realized the complication, my apologies.

To quote myself, apparently I was "sorely mistaken" :S
 
  • #10
jasper10 said:
Integration by substitution surely cannot work!

it's a vicious circle and keeps getting more complicated whilst integrating
Apparently you mean "integration by parts."
 
  • #11
I don't know, nothing seems to work

I'm off to bed now, maybe tomorrow i can think more clearly.
 
  • #12
jasper10 said:
Ok so i get:

∫4x / (x^2 + 1) dx - ∫3/ (x^2 + 1) dx

= 2ln(x^2 + 1) - ?

How would you integrate: 3 / (x^2 + 1)?

the annoying part is the "x^2"

thanks!

ps: what do you mean by "trig" substitution?

(I have a feeling this type of question isn't on the syllabus, but i found it on a past back from 2002 - strange)

[tex]\int \frac{dx}{1 + x^2} = tan^{-1}x + C[/tex]

Presumably you've run across this antiderivative formula and variations of it.
 
  • #13
good morning! :smile:
jasper10 said:
I'm doing my maths baccalaureate in 2 days and came across this question today! …
jasper10 said:
How would you integrate: 3 / (x^2 + 1)?

the annoying part is the "x^2"

thanks!

ps: what do you mean by "trig" substitution?

"trig" substitution is the unofficial but widespread abbreviation for substitution with a trigonometric or hypertrigonometric function (in this case, x = tanu or x = sinhu) :smile:

but you should have known this one anyway, for the exam …

check the PF list of standard integrals and memorise a few :wink:

good luck in your exams! :smile:
 
  • #14
tiny-tim said:
good morning! :smile:



"trig" substitution is the unofficial but widespread abbreviation for substitution with a trigonometric or hypertrigonometric function (in this case, x = tanu or x = sinhu) :smile:

but you should have known this one anyway, for the exam …

check the PF list of standard integrals and memorise a few :wink:

good luck in your exams! :smile:


European baccalaureate or American baccalaureate? Never heard about the term before ?
 
  • #15
European baccalaureate..why? :D
 
  • #16
Ok, thanks Mark44 and tiny-tim!
 
  • #17
jasper10 said:
European baccalaureate..why? :D

Hadn't heard the term before so I googled it. You can check your abilities to integrate

with the Wolfram integrator!
 

1. What does the integral ∫(4x - 3) / (x^2 +1) dx represent?

The integral ∫(4x - 3) / (x^2 +1) dx represents the area under the curve of the function (4x - 3) / (x^2 +1) with respect to the variable x.

2. How do you solve the integral ∫(4x - 3) / (x^2 +1) dx?

To solve this integral, you can use the method of partial fractions to break down the integrand into simpler fractions. Then, you can use the power rule and the natural logarithm rule to integrate each term separately.

3. What is the antiderivative of ∫(4x - 3) / (x^2 +1) dx?

The antiderivative of ∫(4x - 3) / (x^2 +1) dx is (2ln(x^2 + 1) + 4arctan(x)) + C, where C is the constant of integration.

4. Can the integral ∫(4x - 3) / (x^2 +1) dx be evaluated using substitution?

Yes, the integral ∫(4x - 3) / (x^2 +1) dx can be evaluated using the substitution method. By substituting u = x^2 + 1 and du = 2x dx, the integral can be rewritten in terms of u and then evaluated using the power rule and the natural logarithm rule.

5. What is the significance of the constant of integration in the solution to the integral ∫(4x - 3) / (x^2 +1) dx?

The constant of integration represents the sum of all the unknown constants that were lost during the process of integration. It is necessary to include the constant of integration in the solution to account for all possible solutions to the integral.

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