Substitution Rule for Definite Integrals — Question 3

Question 3 - Solution

This integral is well suited for substitution because the derivative of $\ln x$ appears in the denominator.

Let $$u=\ln x.$$ Then $$du=\frac{1}{x}\,dx.$$

Change the limits of integration. When $x=1$, $u=\ln 1=0$. When $x=2$, $u=\ln 2$.

Substitute into the integral: $$\int_{1}^{2} \frac{\ln x}{x}\,dx = \int_{0}^{\ln 2} u\,du.$$

Integrate: $$\int u\,du=\frac{u^{2}}{2}.$$

Apply the limits: $$\frac{u^{2}}{2}\Big|_{0}^{\ln 2} = \frac{(\ln 2)^{2}}{2}.$$

$$\boxed{\frac{(\ln 2)^{2}}{2}}$$