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Question
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Options
`(-1)/"x + 1"` + c
`((-1)/"x + 1")^5` + c
log(x + 1) + c
log |x + 1|5 + c
Solution
`(-1)/"x + 1"` + c
Explanation:
= `int ("x + 1")^3/("x + 1")^5` dx
∵ (a + b)3 = a3 + 3a2b + 3ab2 + b3
(x + 1)3 = x3 + 3x2 + 3x + 1
= `1/((x + 1)^2) dx`
= `int (x + 1)^-2 . dx`
= `(x + 1)^(-2 + 1)/-2 + 1 + c`
= `(x + 1)^-1/-1 + c`
= `(-1)/(x + 1) + c`
`= (-1)/"x + 1"` + c
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