Integrate $\int\frac{1}{x(x^2+1)}dx$ - Techniques & Help

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In summary,In this conversation, the author is trying to find the inverse trig function for sec-2 theta. They first use substitution, but they don't get the correct answer. They then use u substitution and get the correct answer.
  • #1
NotaMathPerson
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I used substitution but did not get a form I know how to integrate. What technique should I use here?
$\int\frac{1}{x(x^2+1)}dx$

Thanks!
 
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  • #2
Perform the substitution

$$\tan\theta=x,\quad\sec^2\theta\,d\theta=dx$$

Can you continue with that?
 
  • #3
greg1313 said:
Perform the substitution

$$\tan\theta=x\quad\sec^2\theta\,d\theta=dx$$

Can you continue with that?

$$\int\frac{\sec^2\theta\,d\theta}{(\tan\theta)(1+\tan^2\theta)}$$

$$\int\frac{\sec^2\theta\,d\theta}{(\tan\theta)(\sec^2\theta)}$$

$$\int\frac{d\theta}{\tan\theta} = \int\frac{\cos\theta\,d\theta}{\sin\theta}$$

Using u substitution

$$\ln|\sin\theta| +c$$ since $$x = \tan\theta = \frac{\sin\theta}{\cos\theta}$$

$$\ln|x\cos\theta| +c$$ is this correct?
 
  • #4
You're correct up to $\ln|\sin\theta|+C$. You need to backsub correctly, so

$$x=\tan\theta,\quad\theta=\tan^{-1}(x)$$

$$\ln|\sin(\tan^{-1}(x))|+C=\ln\left|\dfrac{x}{\sqrt{1+x^2}}\right|+C$$

See this page on inverse trig functions for information on the conversion.
 
  • #5
Can we integrate it using partial fractions?
 
  • #6
NotaMathPerson said:
Can we integrate it using partial fractions?
Of course you can- and it is pretty straight forward. Did you try?

We want to write [tex]\frac{1}{x(x^2+ 1)}= \frac{A}{x}+ \frac{Bx+ C}{x^2+ 1}[/tex].

Eliminate the fractions by multiplying both sides by [tex]x(x^2+ 1)[/tex]:
[tex]1= A(x^2+ 1)+ (Bx+ C)x= Ax^2+ A+ Bx^2+ Cx= (A+ B)x^2+ Cx+ A[/tex].

So, equating "like coefficints", we must have A+ B= 0, C= 0, A= 1. A+ B= 1+ B= 0 so B= -1.

[tex]\int\frac{1}{x(x^2+ 1)}dx= \int \frac{1}{x} dx+ \int\frac{-x}{x^2+ 1}dx[/tex].

The first integral is ln|x|. To integrate the second, let [tex]u= x^2+ 1[/tex].

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NotaMathPerson said:
$$\int\frac{\sec^2\theta\,d\theta}{(\tan\theta)(1+\tan^2\theta)}$$

$$\int\frac{\sec^2\theta\,d\theta}{(\tan\theta)(\sec^2\theta)}$$

$$\int\frac{d\theta}{\tan\theta} = \int\frac{\cos\theta\,d\theta}{\sin\theta}$$

Using u substitution

$$\ln|\sin\theta| +c$$ since $$x = \tan\theta = \frac{\sin\theta}{\cos\theta}$$

$$\ln|x\cos\theta| +c$$ is this correct?
No, of course not. There was no "[tex]\theta[/tex]" in the original integral.
 
  • #7
Note that

$$\int\frac{1}{x(x^2+1)}dx=\frac {1}{2}\int\frac{\frac {-2}{x^3}}{1+\frac {1}{x^2}}dx$$

This approach can be extended to
$$\int\frac{1}{x^p(x^{p+1}+1)}dx$$
 

FAQ: Integrate $\int\frac{1}{x(x^2+1)}dx$ - Techniques & Help

How do I approach solving this integral?

To solve this integral, you can use the technique of partial fractions. First, factor the denominator into (x)(x^2+1). Then, set up the equation A/x + B/(x^2+1) = 1/x(x^2+1) and solve for A and B. Once you have the values for A and B, you can rewrite the integral as A/x + B/(x^2+1) and then integrate each term separately.

Can I use substitution to solve this integral?

While substitution can be used to solve some integrals, it is not the most efficient method for this particular integral. The use of partial fractions is a more straightforward approach.

Is there a specific formula I can use to solve this integral?

There is no specific formula for solving this integral. However, by using the technique of partial fractions, you can break down the integral into smaller, more manageable parts that can be solved individually.

What are some common mistakes to avoid when solving this integral?

Some common mistakes to avoid include forgetting to factor the denominator, not setting up the equation correctly for partial fractions, and making errors while integrating the individual terms. It is always important to double check your work and be mindful of any potential mistakes.

Are there any tips or tricks for solving this integral more efficiently?

One helpful tip is to check if the integral can be simplified further before jumping into the partial fractions technique. Additionally, practicing and becoming familiar with the process of partial fractions can make solving this and similar integrals quicker and easier.

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