7) Use power series to find
R
arctan(x2 )dx.
Solution: This is very similar to the example we had last friday - you can differentiate arctan(x2 ), find the series for the derivative and integrate it to obtain the series for
arctan(x2 ). Then integrate this series term by term to finish the problem.
A faster way is to use the table from the book which tells you
arctan(x) =
∞
X
(−1)n
n=0
arctan(x2 ) =
∞
X
x2n+1
⇒
2n + 1
∞
(−1)n
n=0
x4n+2
(x2 )2n+1 X
=
(−1)n
2n + 1
2n + 1
n=0
Now we integrate
Z
arctan(x2 )dx =
Z X
∞
∞
X
x2n+1
1
(−1)n
dx =
(−1)n
2n + 1
2n + 1
n=0
n=0
1
Z
x4n+2 dx =
∞
X
1
x4n+3
(−1)n
.
2n + 1 4n + 3
n=0
!
+C