Integration by Parts: Int: x*arctan(x) dx

In summary, the person is attempting to complete Calc 2 in 2 months over the summer due to their desire to graduate in 5 years or less. They are currently working on solving an indefinite integral and have attempted to use integration by parts, but have not yet found a solution. They express gratitude for any help in finding the integral.
  • #1
MathHawk
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Because of circumstance (my desire to graduate in 5 years or less), I've been forced to attempt Calc 2 in 2 months time online over the summer. About 75% of it is going smoothly (compared with 105% or so of Calc 1).

Homework Statement



I'm to solve the indefinite integral: [tex]\int[/tex] x * arctan(x) dx



Homework Equations


Integration by parts is done using: [tex]\int[/tex]u dv = uv - [tex]\int[/tex]v du


The Attempt at a Solution


It seems pretty obvious that u = arctan(x) and that dv = x. From this, du = [tex]\frac{1}{1+x^2}[/tex] dx and v = [tex]\frac{1}{2}[/tex]x2.

Using the integration by parts formula:

[tex]\int[/tex] x * arctan(x) dx = [tex]\frac{1}{2}[/tex]x2arctan(x) - [tex]\frac{1}{2}[/tex][tex]\int[/tex][tex]\frac{x^2}{1 + x^2}[/tex]dx



Now integration by parts must be used again. It seems obvious to select u = x2, dv = [tex]\frac{1}{1 + x^2}[/tex]. du = 2xdx, dv = arctan(x).



[tex]\int[/tex] x * arctan(x) dx = [tex]\frac{1}{2}[/tex]x2arctan(x) - [tex]\frac{1}{2}[/tex] ( x2arctan(x) - 2 [tex]\int[/tex]xarctan(x) ).

Simplified:

[tex]\int[/tex] x * arctan(x) dx = 0 + [tex]\int[/tex] x * arctan(x) dx.



While this is very true, it doesn't help me find the integral. Switching my u and dv in either use of the integration by parts formula hasn't yielded a solution for me in my attempts yet. Thank you in advance for your help :smile:.
 
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  • #2
[tex]\frac{x^2}{x^2+1}=\frac{x^2+1 -1}{x^2+1}=1-\frac{1}{x^2+1}[/tex]
 
  • #3
You are my Bokonon, only you tell truths.
 
  • #4
Thank you, in other words.
 

1. What is integration by parts?

Integration by parts is a method used in calculus to evaluate integrals that are in the form of a product of two functions. It involves choosing one function to differentiate and the other to integrate, and then using the product rule to solve the integral.

2. How do I know when to use integration by parts?

Integration by parts is typically used when the integral involves a product of two functions, and there is no obvious substitution that can be made to simplify the integral. It is also useful when the integral involves a function that can be easily integrated but becomes more complicated when differentiated.

3. What is the formula for integration by parts?

The formula for integration by parts is ∫u*dv = uv - ∫v*du, where u and v are the chosen functions, du is the derivative of u, and dv is the integral of v.

4. How do I choose which function to differentiate and which to integrate?

When using integration by parts, a common strategy is to choose the function that becomes simpler when differentiated as the u function, and the more complicated function as the dv function. This will often result in a simpler integral to solve.

5. Can integration by parts be used for definite integrals?

Yes, integration by parts can be used for both indefinite and definite integrals. When using it for definite integrals, the limits of integration should be applied to the entire expression after integrating by parts.

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