Đáp án:
a) $\displaystyle\int\dfrac{\ln x + 3}{x\ln x}dx=\ln x + 3\ln|\ln x| + C$
b) $\displaystyle\int\dfrac{2\ln x}{x(\ln x -4)}dx=2\ln x + 8\ln|\ln x 4| + C$
Giải thích các bước giải:
$\begin{array}{l}a)\quad I = \displaystyle\int\dfrac{\ln x + 3}{x\ln x}dx\\ Đặt\,\,t =\ln x\\ \to dt = \dfrac1xdx\\ \text{Ta được:}\\ \quad I = \displaystyle\int\dfrac{t +3}{t}dt\\ \to I = \displaystyle\int\left(1 + \dfrac3t\right)dt\\ \to I = \displaystyle\int dt + 3\displaystyle\int\dfrac{1}{t}dt\\ \to I = t + 3\ln|t| + C\\ \to I = \ln x + 3\ln|\ln x| + C\\ b)\quad I =\displaystyle\int\dfrac{2\ln x}{x(\ln x -4)}dx\\ Đặt\,\,t = \ln x - 4\\ \to dt = \dfrac1xdx\\ \text{Ta được:}\\ \quad I = 2\displaystyle\int\dfrac{t +4}{t}dt\\ \to I = 2\displaystyle\int\left(1 + \dfrac4t\right)dt\\ \to I = 2\displaystyle\int dt + 8\displaystyle\int \dfrac1tdt\\ \to I =2t + 8\ln|t| + C\\ \to I = 2(\ln x - 4) + 8\ln|\ln x - 4| + C\\ \to I = 2\ln x + 8\ln|\ln x 4| + C \end{array}$
