## Problem Statement

Calculate the definite integral:

$$
\int_{1}^{8} \frac{\ln ^{2} x}{\sqrt[3]{x^{2}}} d x
$$
## Solution

$$
\int_{1}^{8} \frac{\ln ^{2} x}{\sqrt[3]{x^{2}}} d x=
$$

Let:

$$
\begin{aligned}
& u=\ln ^{2} x ; d u=2 \ln x \cdot \frac{1}{x} \cdot d x \\
& d v=\frac{1}{\sqrt[3]{x^{2}}} \cdot d x ; v=3 \sqrt[3]{x}
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.\ln ^{2} x \cdot 3 \sqrt[3]{x}\right|_{1} ^{8}-\int_{1}^{8} 3 \sqrt[3]{x} \cdot 2 \ln x \cdot \frac{1}{x} \cdot d x=\ln ^{2} 8 \cdot 3 \sqrt[3]{8}-\ln ^{2} 1 \cdot 3 \sqrt[3]{1}-6 \cdot \int_{1}^{8} \frac{\ln x}{\sqrt[3]{x^{2}}} d x= \\
& =\ln ^{2} 8 \cdot 6-0 \cdot 3-6 \cdot \int_{1}^{8} \frac{\ln x}{\sqrt[3]{x^{2}}} d x=6 \ln ^{2} 8-6 \cdot \int_{1}^{8} \frac{\ln x}{\sqrt[3]{x^{2}}} d x=
\end{aligned}
$$

Let:

$$
\begin{aligned}
& u=\ln x ; d u=\frac{1}{x} d x \\
& d v=\frac{1}{\sqrt[3]{x^{2}}} \cdot d x ; v=3 \sqrt[3]{x}
\end{aligned}
$$

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

$$
=6 \ln ^{2} 8-6 \cdot\left(\left.\ln x \cdot 3 \sqrt[3]{x}\right|_{1} ^{8}-\int_{1}^{8} 3 \sqrt[3]{x} \cdot \frac{1}{x} d x\right)=
$$

$$
\begin{aligned}
& =6 \ln ^{2} 8-6 \cdot\left(\ln 8 \cdot 3 \sqrt[3]{8}-\ln 1 \cdot 3 \sqrt[3]{1}-3 \cdot \int_{1}^{8} \frac{1}{\sqrt[3]{x^{2}}} d x\right)= \\
& =6 \ln ^{2} 8-6 \cdot\left(\ln 8 \cdot 6-0 \cdot 3-\left.3 \cdot 3 \sqrt[3]{x}\right|_{1} ^{8}\right)= \\
& =6 \ln ^{2} 8-6 \cdot(6 \ln 8-9 \sqrt[3]{8}+9 \sqrt[3]{1})=6 \ln ^{2} 8-6 \cdot(6 \ln 8-18+9)= \\
& =6 \ln ^{2} 8-6 \cdot(6 \ln 8-9)=6 \ln ^{2} 8-36 \ln 8+54
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-23$ » Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

- Last modified: 21:27, 27 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 2-24

## Material from PlusPi