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Question
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Solution
Let I = `int 1/(sqrt(3"x"^2 + 8))` dx
`int 1/(sqrt((sqrt3"x")^2 + (sqrt8)^2))` dx
`= (log |sqrt3"x" + sqrt((sqrt3"x")^2 + (sqrt8)^2)|)/sqrt3` + c
∴ I = `1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 + 8)|` + c
Alternate method:
Let I = `"I" = int 1/sqrt(3"x"^2 + 8) "dx" = 1/sqrt3 int 1/(sqrt ("x"^2 + 8/3)` dx
`= 1/sqrt3 int 1/sqrt("x"^2 + ((2sqrt2)/sqrt3)^2)` dx
`= 1/sqrt3 log |"x" + sqrt("x"^2 + ((2sqrt2)/sqrt3)^2)| + "c"_1`
`= 1/sqrt3 log |"x" + sqrt("x"^2 + 8/3)| + "c"_1`
`= 1/sqrt3 log |(sqrt3"x" + sqrt(3"x"^2 + 8))/sqrt3| + "c"_1`
`= 1/sqrt3 log|sqrt3"x" + sqrt(3"x"^2 + 8)| - 1/sqrt3 log sqrt3 + "c"_1`
∴ I = `1/sqrt3 log |sqrt3"x" + sqrt(3"x"^2 + 8)|` + c
where c = `"c"_1 - 1/sqrt3 log sqrt3`
Notes
The answer in the textbook is incorrect.
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