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Question
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
Options
log 2
2 log 2
`1/2 log 2`
4 log 2
Solution
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to 2 log 2.
Explanation:
Since I = `int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x`
= `int_(-1)^1 x^3/(x^2 + 2|x| + 1) + int_(-1)^1 (|x| + 1)/(x^2 + 2|x| + 1)"d"x`
= `0 + 2 int_0^1 (|x| + 1)/((|x| + 1)^2) "d"x` ....[odd function + even function]
= `2 int_0^1 (x + 1)/(x + 1)^2 "d"x`
= `2 int_0^1 1/(x + 1) "d"x`
= `2|log|x + 1|]_0^1`
= 2 log 2.
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