# Thread: help with a tricky "by parts" integral

1. ## help with a tricky "by parts" integral

I want to integrate x*arcsinx. I let u = arcsinx and dv = x and proceed. When I get to the next step, I have an x^2 numerator and a sqrt(1-x^2) denominator and I am stuck. Could someone help me proceed from here? I used Wolfram Alpha to get an answer and still can't work it out. Thanks!

2. If you have gotten it to

$\displaystyle \int{x\arcsin{x}\,dx} = \frac{x^2\arcsin{x}}{2} - \int{\frac{x^2}{2\sqrt{1 - x^2}}\,dx}$

to evaluate the next integral, let $\displaystyle x = \sin{\theta}$ so that $\displaystyle dx = \cos{\theta}\,d\theta$.

3. Thanks! Just for information's sake, what circumstances alert you to the need for a trig substitution in an integral? I have never really understood the conditions that make that "switch" necessary. Appreciate the help!

4. It just comes with experience, but generally if you have $\displaystyle a^2 \pm x^2$ or $\displaystyle x^2 \pm a^2$ somewhere in the denominator, the plan is to use an appropriate trigonometric or hyperbolic substitution, combined with some form of the Pythagorean Identity to simplify the denominator and to cancel with the derivative.