What is the integral of (x+1)/Square root(4-x^2)?

In summary, the conversation was about evaluating a given integral and discussing different approaches to solving it. One person suggested using partial fractions while another suggested using trigonometric substitution. The conversation ended with one person providing a step-by-step solution using trigonometric substitution.
  • #1
Sympathy
10
0

Homework Statement



Evaluate the following integrals or state that they diverge. Use proper notation.

Integral from 0 to 2 of (x+1)/Square root(4-x^2)



Homework Equations






The Attempt at a Solution



I just substituted x = 2sin(theta) thus dx = 2cos(theta)

I got to the point where it is .5(integral from 0 to 2 of 2tan(theta)) + .5(integral from 0 to 2 of sec(theta)).

I think there's an easier approach at the problem.

suggestions?
 
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  • #2
Partial fractions.
 
  • #3
I think trig subst. is a good way. The limits of integration will change since you went from x to theta.
 
  • #4
So instead of going from 0 to 2, they go from 0 to [itex] \pi/2 [/itex].
 
  • #5
Ya, that's it. Why do people have to take all the fun out of a problem? Why not leave a hint as a hint? This one-up business is boring.
 
  • #6
[tex]\int\frac{x+1}{4-x^2} dx =\frac{3}{4} \int \frac{1}{2-x} dx - \frac{1}{4}\int \frac{1}{2+x} dx[/tex]. Far too simple from there, the only reason I posted even this far was because I wanted to do it as well :)
 
  • #7
Shoot me in the face, i didnt see the sqrt...
 
  • #8
hello friend


ok i will help you as i can


solution

Integral from 0 to 2 of (x+1)/Square root(4-x^2) dx

let x=2sin(&)
dx=2cos(&)d&

=Integral from 0 to [PLAIN]https://www.physicsforums.com/latex_images/12/1253579-0.png [Broken] of (2sin&+1)(2cos&)/2cos& d&

=Integral from 0 to [PLAIN]https://www.physicsforums.com/latex_images/12/1253579-0.png [Broken] of (2sin&+1) d&



i think know it is easy to solve


and i hope that i could help you
 
Last edited by a moderator:

What is trig substitution?

Trig substitution is a mathematical technique used to simplify and solve integrals that contain trigonometric functions.

When should I use trig substitution?

Trig substitution is typically used when an integral contains expressions that cannot be easily integrated using traditional methods, such as polynomials, exponential functions, or logarithmic functions.

What are the steps for solving a trig substitution problem?

The steps for solving a trig substitution problem are:

  1. Identify the appropriate trig substitution to use based on the form of the integral.
  2. Substitute the appropriate trigonometric function into the integral.
  3. Use trigonometric identities to simplify the integral.
  4. Perform any necessary algebraic manipulations.
  5. Integrate the simplified expression.
  6. Substitute back in the original variable if necessary.

What are some common trig substitutions?

Some common trig substitutions include:

  • For √(a² - x²), use x = a sin θ
  • For √(a² + x²), use x = a tan θ
  • For √(x² - a²), use x = a sec θ
  • For √(x² + a²), use x = a cot θ

What are some tips for solving trig substitution problems?

Some tips for solving trig substitution problems include:

  • Always check your answer by differentiating it to make sure it is correct.
  • Be familiar with common trigonometric identities and how to manipulate them.
  • Practice, practice, practice! The more you solve these types of problems, the easier they will become.

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