Đặt \( t = \ln x \):

\(\int \frac{\ln x}{x(2 + \ln x)} \, dx = \int \frac{t}{t + 2} \, dt\)

\(= \int \left( 1 - \frac{2}{t + 2} \right) dt = t - 2 \ln |t + 2| + c.\)

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\( I = \int_1^e \left( x + \frac{2}{x} \ln x \right) \, dx = \left( \frac{x^2}{2} + \ln^2 x \right)_1^e \)

\( = \frac{e^2}{2} + 1 - \frac{1}{2} = \frac{e^2}{2} + \frac{1}{2}. \)

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\( I = \int \frac{e^x}{e^{2x} + 3e^x + 2} \, dx \quad đặt \, t = e^x \)

\( = \int \frac{1}{t^2 + 3t + 2} \, dt = \int \frac{1}{(t+1)(t+2)} \, dt \)

\( = \int \left( \frac{1}{t+1} - \frac{1}{t+2} \right) dt = \ln \left| \frac{t+2}{t+1} \right| + c. \)

Cách 1: Đặt \(t = e^x \)

Cách 2:  
\( \int \frac{1}{e^x + 1} \, dx = \int \frac{1 + e^x - e^x}{1 + e^x} \, dx = \int \left( 1 - \frac{e^x}{1 + e^x} \right) dx \)

\( = x - \ln(1 + e^x) + c. \)

\( = \int \left( e^{2x} - e^x + 1 \right) \, dx \)

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\( \int \frac{1}{e^x - 1} \, dx = \int \frac{1 - e^x + e^x}{e^x - 1} \, dx = \int \left( -1 + \frac{e^x}{e^x - 1} \right) dx \)

\( = -x + \ln|e^x - 1| + c. \)

page48

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Cách 1: Đặt \( t = e^x \).

Cách 2:  
\( \int \frac{e^{2x}}{e^x + 1} \, dx = \int \frac{e^{2x} + e^x - e^x}{e^x + 1} \, dx = \int \left( e^x - \frac{e^x}{e^x + 1} \right) dx \)

\( = e^x - \ln(e^x + 1) + c. \)

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