Integration Strategy — Question 4

Question 4

Identify the hidden derivative and evaluate: $$\int\frac{\ln x}{x}\,dx.$$

Question 4 – Solution

Step 1: Identify the inner function. Since $$\frac{d}{dx}(\ln x)=\frac1x,$$ the factor $dx/x$ is the differential of $\ln x$.

Step 2: Substitute $u=\ln x$. Then $$du=\frac{dx}{x}.$$ Step 3: Rewrite and integrate. $$\begin{align*} \int\frac{\ln x}{x}\,dx &=\int u\,du\\ &=\frac{u^2}{2}+C. \end{align*}$$ Step 4: Return to $x$. $$\boxed{\frac12(\ln x)^2+C}$$