Evaluate: `int(e^(3x))/(4e^(6x)-9)dx`

Explanation

`int frac{e^{3 x} d x}{4 e^{6 x}-9} `

`text { let } e^{3 x}=t `

`Rightarrow e^{3 x} times 3 d x=d t `

`Rightarrow e^{3 x} d x=frac{d t}{3} `

`text { Now, } int frac{e^{3 x} d x}{4 e^{6 x}-9} `

`=frac{1}{3} int frac{d t}{4 t^{2}-9} `

`=frac{1}{3} int frac{d t}{(2 t)^{2}-3^{2}} `

`=frac{1}{3} times frac{1}{2 times 3} log |frac{2 t-3}{2 t+3}| times frac{1}{2}+C `

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