Checkpoint 4.2.
Let \(t \gt 1\text{,}\) compute the following integrals
\begin{equation*}
\int_1^t \frac{1}{x}dx, \int_1^t \frac{1}{x^2}dx,\ \text{and}\
\int_1^t \frac{1}{\sqrt{x}}dx
\end{equation*}
Then decide the convergence of the corresponding improper integrals on \([0,\infty)\text{.}\)
Solution.
\begin{equation*}
\int_1^t \frac{1}{x}dx = \ln|t| - \ln|1| = \ln t.
\end{equation*}
\begin{equation*}
\int_1^t \frac{1}{x^2}dx = \left. -\frac{1}{x} \right|_1^{t} =
\left.\frac{1}{x} \right|_t^1 = 1 - \frac{1}{t}.
\end{equation*}
and
\begin{equation*}
\int_1^t \frac{1}{\sqrt{x}}dx = \left. 2\sqrt{x} \right|_1^{t} =
2\sqrt{t}-2.
\end{equation*}
Therefore, as \(t \to +\infty\text{,}\) we have
\begin{equation*}
\int_1^{\infty} \frac{1}{x} dx = +\infty, \int_1^{\infty} \frac{1}{x^2} dx=
1, \int_1^{\infty}\frac{1}{\sqrt{x}} dx = +\infty.
\end{equation*}