SOLUTION: integral of x cos^-1 x dx
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Question 697886: integral of x cos^-1 x dx
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
\,dx})
Use Integration by Parts:
Let )
Then from any standard table of derivatives: 
So 
And 
So 
Then substituting into the Integration by Parts formula:
\,dx}\ =\ \frac{1}{2}x^2\,\arccos(x)\ +\ \frac{1}{2}\int\,\frac{x^2}{\sqrt{1\,-\,x^2}}\,dx)
To calculate the remaining integral, let 
Then 
Then 
And then 
But since 
,
Rewrite: )
\,d\varphi)
Then since 
, )
and 
, so:
\ -\ \frac{1}{2}x\sqrt{1\,-\,x^2}\ +\ C)
Finally, putting the pieces back together:
\,dx}\ =\ \frac{1}{2}x^2\arccos(x)\ +\ \frac{1}{4}\arcsin(x)\ -\ \frac{1}{4}x\sqrt{1\,-\,x^2}\ +\ C)
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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