Problem source

Find the following integrals:
\begin{questionparts}
\item $\ {\displaystyle \int_{1}^{\mathrm{e}}\frac{\ln x}{x^{2}}\,\mathrm{d}x}\,,$
\item $\ {\displaystyle \int\frac{\cos x}{\sin x\sqrt{1+\sin x}}\,\mathrm{d}x.}$
\end{questionparts}

Solution source

\begin{questionparts}
\item \begin{align*}
\int_{1}^{\mathrm{e}}\frac{\ln x}{x^{2}}\,\mathrm{d}x &= \left [-\frac{\ln x}{x} \right]_1^e + \int_1^e \frac{1}{x^2} \, \d x \\
&= -\frac{1}{e} + \left [ -\frac{1}{x} \right]_1^e \\
&= 1 - \frac{2}{e}
\end{align*}

\item 
\begin{align*}
\int\frac{\cos x}{\sin x\sqrt{1+\sin x}}\,\mathrm{d}x &= \int \frac{2u}{(u^2-1)u} \d u \tag{$u^2 = 1+\sin x$} \\
&= \int \frac{1}{u-1} - \frac{1}{u+1} \d u \\
&= \ln(u-1) - \ln (u+1) + C \\
&= \ln \l \frac{u-1}{u+1} \r + C \\
&= \ln \l \frac{\sqrt{\sin x + 1} + 1}{\sqrt{\sin x + 1} -1} \r + C
\end{align*}
\end{questionparts}