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Question
`int x^3/(x + 1)` is equal to ______.
Options
`x + x^2/2 + x^3/3 - log|1 - x| + "C"`
`x + x^2/2 - x^3/3 - log|1 - x| + "C"`
`x - x^2/2 - x^3/3 - log|1 + x| + "C"`
`x - x^2/2 + x^3/3 - log|1 + x| + "C"`
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Solution
`int x^3/(x + 1)` is equal to `x - x^2/2 + x^3/3 - log|1 + x| + "C"`.
Explanation:
I = `int x^3/(x + 1)`
= `int (x^3 + 1 - 1)/(x + 1) "d"x`
= `int (x^3 + 1)/(x + 1) "d"x - int 1/(x + 1) "d"x`
= `int (x^2 - x + 1)"d"x - int 1/(x + 1) "d"x`
= `x^3/3 - x^2/2 + x - log|x + 1| + "C"`
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