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Question
Integrate the following with respect to x.
`"e"^(3x) [(3x - 1)/(9x^2)]`
Solution
`"e"^(3x) [(3x - 1)/(9x^2)] = "e"^(3x) [1/(3x) - 1/(9x^2)]`
= `"e"^(3x)/9 [3(1/x) + (- 1)/x^2]`
Let f(x) = `1/x`
∴ f'(x) = `(- 1)/x^2`
So `int "e"^(3x) [(3x - 1)/(9x^2)] "d"x = 1/9 int "e"^(3x) [3(1/x) - 1/x^2] "d"x`
= `1/9 int "e"^("a"x) ["a" "f"(x) + "f'"(x)] "d"x`
= `1/9 ["e"^("a"x) "f"(x)] + "c"`
= `1/9 ["e"^(3x) (1/x)] + "c"`
= `"e"^(3x)/(9x)+ "c"`
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