## Problem Statement

Calculate the definite integral:

$$
\int_{1}^{e^{2}} \frac{\ln ^{2} x}{\sqrt{x}} d x
$$
## Solution

$$
\int_{1}^{\epsilon^{2}} \frac{\ln ^{2} x}{\sqrt{x}} d x=
$$

Let:

$$
\begin{aligned}
& u=\ln ^{2} x ; d u=2 \ln x \cdot \frac{1}{x} \cdot d x \\
& d v=\frac{1}{\sqrt{x}} \cdot d x ; v=2 \sqrt{x}
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =\left.\ln ^{2} x \cdot 2 \sqrt{x}\right|_{1} ^{e^{2}}-\int_{1}^{e^{2}} 2 \sqrt{x} \cdot 2 \ln x \cdot \frac{1}{x} \cdot d x=\ln ^{2}\left(e^{2}\right) \cdot 2 \sqrt{e^{2}}-\ln ^{2} 1 \cdot 2 \sqrt{1}-4 \int_{1}^{e^{2}} \frac{\ln x}{\sqrt{x}} \cdot d x= \\
& =2^{2} \cdot 2 e-0 \cdot 2 \sqrt{1}-4 \int_{1}^{e^{2}} \frac{\ln x}{\sqrt{x}} \cdot d x=8 e-4 \int_{1}^{e^{2}} \frac{\ln x}{\sqrt{x}} \cdot d x=
\end{aligned}
$$

Let:

$$
\begin{aligned}
& u=\ln x ; d u=\frac{1}{x} d x \\
& d v=\frac{1}{\sqrt{x}} \cdot d x ; v=2 \sqrt{x}
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =8 e-4 \cdot\left(\left.\ln x \cdot 2 \sqrt{x}\right|_{1} ^{e^{2}}-\int_{1}^{\epsilon^{2}} 2 \sqrt{x} \cdot \frac{1}{x} d x\right)= \\
& =8 e-4 \cdot\left(\ln \left(e^{2}\right) \cdot 2 \sqrt{e^{2}}-\ln 1 \cdot 2 \sqrt{1}-2 \int_{1}^{e^{2}} \frac{d x}{\sqrt{x}}\right)=8 e-4 \cdot\left(2 \cdot 2 e-0-\left.2 \cdot 2 \sqrt{x}\right|_{1} ^{e^{2}}\right)=
\end{aligned}
$$

$$
=8 e-4 \cdot\left(4 e-4 \sqrt{e^{2}}+4 \sqrt{1}\right)=8 e-4 \cdot(4 e-4 e+4)=8 e-16
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� $\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D0} \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{H} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D1} \% 82$ $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} 2-22$ "

Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

- Last modified: 21:01, 27 February 2009.
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## Problem Kuznetsov Integrals 2-23

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