Question

$\int \log 2xdx$ is equal to

• A. $x\log 2x-\frac{x^2}{2}+c$
• B. $x\log 2x-\frac{x}{2}+c$
• C. $x^2\log 2x-\frac{x}{2}+c$
• D. $x\log 2x-x+c$

Answer 1

Answer: (D)

Let $I=\int \log 2xdx$ or $I=\int 1\cdot \log 2xdx\Rightarrow$ \begin{array}{l} I=\log 2 x \int 1 \cdot d x-\int\left\{\left(\frac{d}{d x} \log 2 x\right) \int 1 d x\right) d x \Rightarrow \\ I=x \log 2 x-\int \frac{1}{2 x} \cdot 2 \cdot x {\quad} d x \Rightarrow I=x \log 2 x-x+c \end{array}