Mathematics - Indefinite Integration Question with Solution | TestHub

Differentiate both sides with respect to $x$:

$$\begin{array}{rl}\frac{1}{(1+\log x)(2+\log x)(3+\log x)x}& =\left(\frac{A}{1+\log x}+\frac{B}{2+\log x}+\frac{C}{3+\log x}\right)\frac{1}{x}\\ 1& =A(2+t)(3+t)+B(1+t)(3+t)+C(1+t)(2+t),t=\log x\\ Substitutet=-2& \Rightarrow B=-1\\ Substitutet=-3& \Rightarrow C=\frac{1}{2}\\ Substitutet=-1& \Rightarrow A=\frac{1}{2}\\ \Rightarrow A-B+C& =2\end{array}$$