Solved: Help with Primitive Calculus Problem

  • Thread starter bnsm
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In summary, the conversation is about a user seeking help with solving the integral \int\frac{dx}{x^{2}\sqrt{4-x^{2}}}. They have tried substitution and integration by parts, but are stuck on how to proceed. Another user suggests using trig substitution and shows them how to simplify the integral to \int\frac{dt}{4sin^{2}\left(t\right)}. The first user then tries to use integration by parts again, but the second user points out that the derivative of cot(t) is -1/sin(t)^2, making the integral much simpler.
  • #1
bnsm
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Help in a primitive!

Homework Statement



Hello guys! Please, I'm really needing help in a primitive... I don't know, maybe it has a simple solution, but I'm tired and blocked on this... Can you give some lights? Here goes the equation:

[tex]\int\frac{dx}{x^{2}\sqrt{4-x^{2}}}[/tex]

Homework Equations





The Attempt at a Solution



I tried substitution of 4-x^2 and of x^2, but none of them work... I also tried by parts, with u'=1/(x^2) and v=1/sqrt(4-x^2), but it looks like it becomes even heavier... Can you help me?

Thanks to all and to this great site!
 
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  • #2


Trig substitution is the obvious best choice here.
 
  • #3


Tom Mattson said:
Trig substitution is the obvious best choice here.

Yes, of course, you're right! Many Thanks! :) I made x=2*sin(t) and I got:

[tex]\int\frac{dt}{4sin^{2}\left(t\right)}[/tex]
 
  • #4


Ok, I'm stucked again... I tried:

[tex]\frac{1}{4}\int\frac{sin^{2}\left(t\right)+cos^{2}\left(t\right)}{sin^{2}\left(t\right)}dt[/tex]

which gave:

[tex]\frac{t}{4}+\int\frac{cos^{2}\left(t\right)}{sin^{2}\left(t\right)}dt[/tex]

Any ideas? I tried partial and substitution but it's a mess...
 
  • #5


Try and differentiate cot(x)=cos(x)/sin(x), ok? What do you get?
 
  • #6


Dick said:
Try and differentiate cot(x)=cos(x)/sin(x), ok? What do you get?

I substituted the fraction above by the cot(t) and then I made the primitive by parts, considering

u'=1 and thus u=t
v=cot(t) and thus v'=-2cot(t)/((sin(t))^2)

Then, I tried to develop the following:

[tex]\int\frac{cos^{2}\left(t\right)}{sin^{2}\left(t\right)}=t\cot^{2}\left(t\right)+\int\frac{2t\cot\left(t\right)}{sin^{2}\left(t\right)}[/tex]

What do you think about this? I can try to substitute cot(t) by cos(t)/sin(t), but I'll get a (sin(x))^3 in the denominator... The point is that it seems I'm getting a primitive even more complicated...
 
  • #7


You are making this way too complicated. You wanted to find the integral of dt/sin(t)^2. All I was trying to point out is that the derivative of cot(t) is -1/sin(t)^2. Doesn't that make it easy?
 

Related to Solved: Help with Primitive Calculus Problem

1. What is primitive calculus?

Primitive calculus, also known as integral calculus, is a branch of mathematics that deals with finding the area under a curve, and is the inverse operation of derivative calculus. It involves the calculation of indefinite and definite integrals, as well as techniques for solving integration problems.

2. What is a primitive calculus problem?

A primitive calculus problem is a mathematical question that involves finding the antiderivative or integral of a given function. This typically involves using techniques such as substitution, integration by parts, or other methods to find the solution.

3. How do I solve a primitive calculus problem?

To solve a primitive calculus problem, you first need to identify the function and determine the appropriate technique to use. Next, you will integrate the function and apply any necessary rules or properties to simplify the solution. Finally, you will evaluate the solution to get the final answer.

4. What are some common techniques used to solve primitive calculus problems?

Some common techniques used to solve primitive calculus problems include substitution, integration by parts, partial fractions, and trigonometric substitutions. It is important to have a good understanding of these techniques and when to use them in order to successfully solve integration problems.

5. How can I improve my skills in solving primitive calculus problems?

The best way to improve your skills in solving primitive calculus problems is to practice regularly and familiarize yourself with the various integration techniques. You can also seek help from textbooks, online resources, or a tutor to gain a better understanding of the concepts and methods involved in solving integration problems.

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