How to evaluate this indefinite integral?

In summary, the conversation discusses the process of solving the integral \int \frac{\sqrt{x^4-1}}{x^3}dx and the various methods attempted. The conversation concludes with the correct solution, which involves using the quadratic formula and substitution to get \int \sec \theta d\theta - \int \cos \theta d\theta = \ln |x^2+\frac{\sqrt{1-x^2}}{x^2}|-\sqrt{1-x^2}+C.
  • #1
shakgoku
29
1

Homework Statement



[tex]\int \frac{\sqrt{x^4-1}}{x^3}dx[/tex]

Homework Equations





The Attempt at a Solution



tried substituting [tex]x^4 = \sec^2 \theta[/tex] to get rid of square root but it was of no use because, I got another complex integral [tex]\int \frac{\sin^2\theta}{\cos^{\frac{3}{4}} \theta}d\theta[/tex]

Is there an easy way to solve this problem?
 
Last edited:
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  • #2
why can't u take x^3 inside square root. Then it will be simplified to
sqrt(1-x^2)/x which is easier to integrate
 
  • #3
test!
The quadratic formula is
http://texify.com/$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ [Broken]
 
Last edited by a moderator:
  • #4
n.karthick said:
why can't u take x^3 inside square root. Then it will be simplified to
sqrt(1-x^2)/x which is easier to integrate

I don't get it. if we take x^3 inside we get,

[tex]\int \sqrt{\frac{x^4-1}{x^6}}dx=\int \sqrt{\frac{1}{x^2}-\frac{1}{x^6}}dx[/tex]
 
  • #5
I just realized I made a blunder in calculation.
That substitution yields
[tex]\int \frac{\sin^2\theta}{cos \theta}d \theta=\int \frac{1-\cos^2\theta}{\cos \theta} d\theta[/tex]

and finally it is equal to,

[tex]\int \sec \theta d\theta-\int \cos\theta d\theta = \ln |x^2+\frac{\sqrt{1-x^2}}{x^2}|-\sqrt{1-x^2}+C[/tex]

Which is the answer!
:peace:
 

1. How do I determine the limits of integration?

The limits of integration are typically given in the problem or can be identified by looking at the function being integrated. If the function is continuous, the limits will be the points at which the function changes behavior.

2. What is the best method for evaluating indefinite integrals?

The best method for evaluating indefinite integrals depends on the specific problem at hand. Some common methods include substitution, integration by parts, and partial fractions. It is important to carefully examine the integral and choose the most appropriate method for solving it.

3. Can I use a calculator to evaluate indefinite integrals?

Yes, many calculators have the capability to evaluate indefinite integrals. However, it is still important to understand the steps involved in solving the integral by hand in order to check the accuracy of the calculator's result.

4. How do I know if my answer is correct?

You can check your answer by taking the derivative of the result. If the derivative matches the original function, then your answer is correct. It is also helpful to use online tools or a graphing calculator to visualize the function and its integral to verify the accuracy of your answer.

5. Can I use a different variable for my integration?

Yes, in most cases you can use any variable for integration as long as it is properly defined in the problem. However, it is common to use x as the variable for indefinite integrals.

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