Finding the Anti-Derivative of (2+x^2)/(1+x^2): A Scientific Approach

  • Thread starter joeG215
  • Start date
  • Tags
    Derivative
In summary, the conversation is about finding the anti derivative of f'(x)=(2+x^2)/(1+x^2). The attempted solution involves simplifying the expression using inverse tan derivative and long division, but the individual is unsure of how to proceed. Suggestions are made to use substitution and dividing out fractions, and the final solution is given as inverse tan of x + x + C.
  • #1
joeG215
18
0

Homework Statement



f'(x)= (2+x^2)/(1+x^2) Find anti derivative

Homework Equations


The Attempt at a Solution



I attempted to bring the denominator up using (1+x^2)^-1 and i also tried long division to simplify but had no luck...

1/(1+x^2) is the inverse tan derivative, but what can i do from here:

(2+x^2) * 1/(1+x^2) is substitution legal here?
 
Last edited:
Physics news on Phys.org
  • #2
Try putting (2+x^2)/(1+x^2) as 2/(1+x^2) + x^2/(1+x^2)

then divide out the second fraction
 
  • #3
Or write
[tex]\frac{2+x^2}{1+ x^2}= \frac{1}{1+x^2}+ \frac{1+ x^2}{1+ x^2}[/tex]
 
  • #4
so i would cancel the second term then take the anti derivate to be left with invesre tan of x + x + C .. is this correct?
 

1. What is an anti derivative problem?

An anti derivative problem is a mathematical problem that involves finding the original function whose derivative is a given function. It is the reverse process of differentiation.

2. How do you solve an anti derivative problem?

To solve an anti derivative problem, you can use integration techniques such as substitution, integration by parts, or partial fractions. You can also use tables of common derivatives and integrals to help identify the original function.

3. What is the difference between an anti derivative and an indefinite integral?

An indefinite integral is a family of functions that differ by a constant, while an anti derivative is a specific function that gives the same derivative as the given function. In other words, an anti derivative is a specific solution to an indefinite integral.

4. Can all functions have an anti derivative?

No, not all functions have an anti derivative. For a function to have an anti derivative, it must be continuous and have a finite number of discontinuities. Functions that do not meet these criteria, such as the Dirichlet function, do not have an anti derivative.

5. How are anti derivatives used in real life?

Anti derivatives have many applications in real life, particularly in physics and engineering. They are used to calculate the total distance traveled by an object given its velocity, or to find the total amount of work done by a variable force. They are also used in economics to model the growth or decay of a population or investment.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
505
  • Calculus and Beyond Homework Help
Replies
2
Views
459
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
916
  • Calculus and Beyond Homework Help
2
Replies
44
Views
3K
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
803
Back
Top