Монгол Бодлогын Сан

Заавар: $\displaystyle\int_1^{e^4}\dfrac{\,\mathrm{d}x}{x(\ln x+1)}=\int_1^{e^4}\dfrac{\mathrm{d}(\ln x+1)}{\ln x+1}$ байна.

Бодолт: \begin{gather} \int_1^{e^4}\dfrac{\,\mathrm{d}x}{x(\ln x+1)}=\int_1^{e^4}\dfrac{\mathrm{d}(\ln x+1)}{\ln x+1}\\ =\int_{\ln 1+1}^{\ln e^4+1}\dfrac{\mathrm{d}t}{t}=\int_1^5\dfrac{\mathrm{d}t}{t}=\ln t\bigg|_1^5=\ln5 \end{gather}