Antiderivatives and definite integrals Compute the following integral: I = <msubsup>

2nalfq8 2022-07-06 Answered
Antiderivatives and definite integrals
Compute the following integral:
I = 0 1 ( 1 x 1 1 + t 2 d t ) d x
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Answers (2)

Maggie Bowman
Answered 2022-07-07 Author has 14 answers
Explanation:
0 1 1 x d t 1 + t 2 d x = 0 1 ( arctan ( x ) π 4 ) d x = [ x arctan ( x ) 1 2 ln ( 1 + x 2 ) π 4 x ] 0 1 = 1 2 ln ( 2 )
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dream13rxs
Answered 2022-07-08 Author has 4 answers
Step 1
Let f ( t ) = 1 1 + t 2 , t R and F ( x ) = 1 x 1 1 + t 2 d t = 1 x f ( t ) d t, t R
Step 2
Consequently, F ( x ) = f ( x ), x R and:
I = 0 1 ( 1 x 1 1 + t 2 d t ) d x = 0 1 F ( x ) d x = 0 1 ( x ) F ( x ) d x =
= [ x F ( x ) ] 0 1 0 1 x F ( x ) d x = F ( 1 ) 0 1 x f ( x ) d x = 0 0 1 x 1 + x 2 d x =
= 1 2 0 1 ( 1 + x 2 ) 1 + x 2 d x = 1 2 [ ln ( 1 + x 2 ) ] 0 1 = ln 2 2
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