MathHawk
May27-09, 10:28 PM
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).
1. The problem statement, all variables and given/known data
I'm to solve the indefinite integral: \int x * arctan(x) dx
2. Relevant equations
Integration by parts is done using: \intu dv = uv - \intv du
3. The attempt at a solution
It seems pretty obvious that u = arctan(x) and that dv = x. From this, du = \frac{1}{1+x^2} dx and v = \frac{1}{2}x2.
Using the integration by parts formula:
\int x * arctan(x) dx = \frac{1}{2}x2arctan(x) - \frac{1}{2}\int\frac{x^2}{1 + x^2}dx
Now integration by parts must be used again. It seems obvious to select u = x2, dv = \frac{1}{1 + x^2}. du = 2xdx, dv = arctan(x).
\int x * arctan(x) dx = \frac{1}{2}x2arctan(x) - \frac{1}{2} ( x2arctan(x) - 2 \intxarctan(x) ).
Simplified:
\int x * arctan(x) dx = 0 + \int 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:.
1. The problem statement, all variables and given/known data
I'm to solve the indefinite integral: \int x * arctan(x) dx
2. Relevant equations
Integration by parts is done using: \intu dv = uv - \intv du
3. The attempt at a solution
It seems pretty obvious that u = arctan(x) and that dv = x. From this, du = \frac{1}{1+x^2} dx and v = \frac{1}{2}x2.
Using the integration by parts formula:
\int x * arctan(x) dx = \frac{1}{2}x2arctan(x) - \frac{1}{2}\int\frac{x^2}{1 + x^2}dx
Now integration by parts must be used again. It seems obvious to select u = x2, dv = \frac{1}{1 + x^2}. du = 2xdx, dv = arctan(x).
\int x * arctan(x) dx = \frac{1}{2}x2arctan(x) - \frac{1}{2} ( x2arctan(x) - 2 \intxarctan(x) ).
Simplified:
\int x * arctan(x) dx = 0 + \int 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:.