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Question
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Solution
Let I = `int "dx"/(4"x"^2 - 1)`
`= 1/4 int "dx"/("x"^2 - 1/4)`
`= 1/4 int "dx"/("x"^2 - (1/2)^2)`
`= 1/4 xx 1/(2 xx 1/2) log |("x" - 1/2)/("x" + 1/2)|` + c
∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c
Alternate Method:
Let I = `int "dx"/(4"x"^2 - 1) = int "dx"/((2"x"^2) - (1)^2)`
`= 1/(2 xx 1) xx 1/2 log |("2x" - 1)/("2x" + 1)|` + c
∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c
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