An integration involving inverse trig.

Another user helps them by suggesting to split up the fraction and use a substitution and inverse tangent to solve the problem. Charismaztex thanks the user for their help. In summary, Charismaztex asks for help with integrating a complex expression and another user suggests a method involving splitting up the fraction and using a substitution and inverse tangent. Charismaztex expresses gratitude for the help.
  • #1
Charismaztex
45
0

Homework Statement



The question is integrate

[tex]\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx[/tex]

Homework Equations



[tex]\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx =2 + ln(\frac{5}{2}) +\frac{1}{2}arctan(\frac{1}{2})[/tex] (yes, it's a prove question)

The Attempt at a Solution



The first two parts I have no problem but I am not quite sure how to integrate the

[tex]\frac{2x+1}{x^2+4}[/tex] part. I know it has to do with arctan but I'm not quite sure about the [tex]2x+1[/tex] part.

Thanks in advance,
Charismaztex
 
Physics news on Phys.org
  • #2
Split up the last fraction so that you have [tex]\frac{2x}{x^{2}+4} + \frac{1}{x^{2}+4}[/tex]. Use a simple substitution for the first fraction and then the second fraction will involve the inverse tangent. In general,

[tex]\int \frac{dx}{x^2+a^2} = \frac{1}{a} arctan(\frac{x}{a}) + C[/tex]
 
  • #3
ahh! I completely missed that step! Thanks for your help :)

Charismaztex
 

1. What is an integration involving inverse trig?

An integration involving inverse trig refers to the process of finding the antiderivative of a function that contains inverse trigonometric functions, such as arcsine, arccosine, or arctangent. This type of integration is often used to solve trigonometric equations and evaluate definite integrals.

2. What is the general formula for integrating inverse trig functions?

The general formula for integrating inverse trig functions is: ∫f(x)dx = x⋅arctan(x) + ½ln(1+x^2) + C, where C is the constant of integration. This formula can be used for integrating any inverse trig function, such as arcsine, arccosine, or arctangent.

3. What are some common techniques for solving integrals involving inverse trig?

Some common techniques for solving integrals involving inverse trig include using substitution, integration by parts, and trigonometric identities. It is also helpful to have a good understanding of the derivatives of inverse trig functions.

4. Are there any special cases when integrating inverse trig functions?

Yes, there are a few special cases when integrating inverse trig functions. For example, when integrating arctangent, the substitution u = tan(x/2) can be used to simplify the integral. When integrating arccosine, the substitution u = sin(x) can be used. Additionally, when integrating arcsine, the substitution u = cos(x) can be used.

5. How can integrals involving inverse trig be applied in real life?

Integrals involving inverse trig can be used in various fields of science and engineering, such as physics, astronomy, and electrical engineering. For example, in physics, these integrals can be used to calculate the displacement, velocity, and acceleration of objects moving in circular or oscillatory motion. In astronomy, they can be used to calculate the positions and movements of celestial objects. In electrical engineering, they can be used to analyze and design circuits involving alternating current.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
283
  • Calculus and Beyond Homework Help
Replies
2
Views
479
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
Replies
5
Views
934
  • Calculus and Beyond Homework Help
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
278
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
719
  • Calculus and Beyond Homework Help
Replies
3
Views
269
  • Calculus and Beyond Homework Help
Replies
12
Views
935
Back
Top