Need a find int_(.5)^2 ln(4+x^2)dx

Ashleigh Clarke 2022-10-20 Answered
Need a find .5 2 l n ( 4 + x 2 ) d x I understand that to find this value I need to find what the power series of l n ( 4 + x 2 ) and I know how to start this off since ln ( 1 x ) = 1 1 x which is 1 1 x is the geometric series.
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Answers (1)

zupa1z
Answered 2022-10-21 Author has 20 answers
Starting with ln ( 1 + x ) = n = 1 ( 1 ) n x n n (for |x|<1), observe that
ln ( 4 + x 2 ) = ln 4 + ln ( 1 + x 2 4 ) = 2 ln 2 + ln ( 1 + x 2 4 ) .
In particular, we can apply the result above to ln ( 1 + y ), with y = x 2 4 (note that 0 x < 2 implies 0 y < 1):
ln ( 4 + x 2 ) = 2 ln 2 n = 1 ( 1 ) n n 2 2 n x 2 n .
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