Đặt:
\( \begin{cases}
u = 1 + \ln(x+1) \\
dv = \frac{1}{x^2} \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{1}{x+1} \, dx \\
v = -\frac{1}{x}.
\end{cases} \)
Khi đó:
\( I = -\frac{1}{x} (1 + \ln(x+1)) + \int \frac{1}{x(x+1)} \, dx \)
\( = -\frac{1}{x} (1 + \ln(x+1)) + \int \left( \frac{1}{x} - \frac{1}{x+1} \right) \, dx \)
\( = -\frac{1}{x} (1 + \ln(x+1)) + \ln \left| \frac{x}{x+1} \right| + C. \)
page80
Đặt: \( \begin{cases}
u = \ln x \\
dv = \frac{x^2 - 1}{x^2} \, dx = \left(1 - \frac{1}{x^2}\right) \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{1}{x} \, dx \\
v = x + \frac{1}{x}.
\end{cases} \)
\( I = \left(x + \frac{1}{x}\right) \ln x - \int \left(x + \frac{1}{x}\right) \frac{1}{x} \, dx \)
\(= \left(x + \frac{1}{x}\right) \ln x - \int \left(1 + \frac{1}{x^2}\right) \, dx \)
\(= \left(x + \frac{1}{x}\right) \ln x - \left(x - \frac{1}{x}\right) + C. \)
\( \begin{cases}
u = 3 + \ln x \\
dv = \frac{1}{(x+1)^2} \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{1}{x} \, dx \\
v = -\frac{1}{x+1}.
\end{cases} \)
page81
Đặt: \( \begin{cases}
u = \ln(x^2 - x) \\
dv = dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{2x - 1}{x^2 - x} \, dx \\
v = x.
\end{cases} \)
\( I = x \ln(x^2 - x) - \int \frac{2x - 1}{x-1} \, dx \)
\( I = x \ln(x^2 - x) - \int \left( 2 + \frac{1}{x - 1} \right) \, dx \)
\( I = x \ln(x^2 - x) - 2x - \ln|x - 1| + C. \)
page82
Đặt: \( \begin{cases}
u = \ln(\sin x) \\
dv = \frac{1}{\cos^2 x} \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{\cos x}{\sin x} \, dx \\
v = \tan x.
\end{cases} \)
\( I = \tan x \cdot \ln(\sin x) - \int dx \)
\( = \tan x \cdot \ln(\sin x) - x + C. \)
page83
Đặt: \( \begin{cases}
u = \ln(1 + \cos x) \\
dv = \cos x \, dx
\end{cases} \quad \Rightarrow \quad
\begin{cases}
du = \frac{-\sin x}{1 + \cos x} \, dx \\
v = \sin x.
\end{cases} \)
\( I = \sin x \ln(1 + \cos x) + \int \frac{\sin^2 x}{1 + \cos x} \, dx \)
\(= \sin x \ln(1 + \cos x) + \int (1 - \cos x) \, dx \)
\( = \sin x \ln(1 + \cos x) + x - \sin x + C. \)
page84