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in Indefinite Integral by (65.2k points)

Evaluate: \(\int\frac{3e^x}{e^{2x} + 5e^x + 6}dx\)

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\(\int\frac{3e^x}{e^{2x} + 5e^x + 6}dx\)

= 3\(\int\frac{t}{t^2 + 5t + 6}dt\)

put ex = t

= 1\(\int\frac{3tdt}{(t + 2)(t + 3)} = \int\frac{A}{t + 2} + \frac{B}{t + 3}dt\)

Let 36 = A(t + 3) + B(t + 2) 

put t = -2 - 6 = A(1) ⇒ A = -6 

put t = -3, - 9 = A(0) + B(-1) ⇒ B = 9

∴ \(\int\frac{3e^x}{(e^x)^2 + 5e^x + 6} = \int\frac{-6}{t + 2} + \frac{9}{t + 3}dt\)

= -6log(t + 2) + 9log(t + 3) + C = -6log(ex + 3) + 9 log(ex + 3) + c

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