Integral of x arctan x dx

In summary, the conversation discusses the solution to the integral \int x \arctan x \, dx using the by parts method. The solution involves using the substitution u=x^2+1 to simplify the process. The conversation also discusses adding and subtracting 1 from the numerator and ends with a helpful resource for further understanding.
  • #1
alba_ei
39
1

Homework Statement


[tex] \int x \arctan x \, dx [/tex]

The Attempt at a Solution


By parts,
[tex] u = \arctan x[/tex]
[tex] dv = x dx[/tex]
[tex] du = \frac{dx}{x^2+1}[/tex]
[tex]v = \frac{x^2}{2} [/tex]

[tex] \int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{1}{2} \int \frac{x^2}{x^2+1} \, dx [/tex]

Again...by parts

[tex] u = x^2 [/tex]
[tex] dv = \frac{dx}{x^2+1} [/tex]
[tex] du = 2x dx [/tex]
[tex] v = arc tan x [/tex]

[tex]\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx [/tex]
I back to the beginning, what did wrogn?

[tex]\int x \arctan x \, dx = - \int x \arctan x \, dx [/tex]
 
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  • #2
[tex]\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx [/tex]

Add [tex]\int x \arctan x \, dx[/tex] to both sides, then solve for the integral, assuming your work is correct.
 
  • #3
z-component said:
[tex]\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx [/tex]

Add [tex]\int x \arctan x \, dx[/tex] to both sides, then solve for the integral, assuming your work is correct.

you mean like this? is the same, i back to the beginign

[tex]\int x \arctan x \, dx +\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx +\int x \arctan x \, dx[/tex]

[tex]2\int x \arctan x \, dx = 0[/tex]
 
  • #4
alba_ei said:
[tex]- \frac{1}{2} \int \frac{x^2}{x^2+1} \, dx [/tex]
Why use 'by parts' again? It would easier if you just add and subtract 1 from the numerator
 
  • #5
why not try the substitution u=x^2+1 in that second integral...
 
  • #6
for the integral x²/(x²+1)
you can rewrite it as (x² + 1 - 1)/(x²+1) => 1 - 1/(x²+1)
 
  • #7
umm hmm, that leaves a nice (x - arctan x) for you there.
 
  • #8
http://www.maths.abdn.ac.uk/~igc/tch/ma1002/int/node34.html
Example 3.15
 
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What is the integral of x arctan x dx?

The integral of x arctan x dx is equal to (x^2/2)(arctan x) - (x^2/4)ln(|x^2 + 1|) + C, where C is the constant of integration.

How do you solve the integral of x arctan x dx?

To solve the integral of x arctan x dx, you can use integration by parts or substitution. Integration by parts involves breaking down the integral into two parts and using a formula to find the answer. Substitution involves replacing the variable with a new one and then solving the integral using integration rules.

What is the purpose of solving the integral of x arctan x dx?

Solving the integral of x arctan x dx allows us to find the area under the curve of the function. This can be useful in many applications, such as calculating work done in physics or finding the average value of a function.

What is the domain of the integral of x arctan x dx?

The domain of the integral of x arctan x dx is [-∞, ∞]. This means that the integral can be evaluated for any real value of x.

What are some real-life applications of the integral of x arctan x dx?

The integral of x arctan x dx has many applications in physics, engineering, and economics. It can be used to find the work done by a variable force, the deflection of a beam under a variable load, or the average cost of production for a company. It can also be used in statistics to calculate the average of a data set or in signal processing to find the center frequency of a signal.

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