## Problem Statement

Calculate the definite integral:

$$
\int_{-5 / 3}^{1} \frac{\sqrt[3]{3 x+5}+2}{1+\sqrt[3]{3 x+5}} d x
$$
## Solution

$$
\int_{-5 / 3}^{1} \frac{\sqrt[3]{3 x+5}+2}{1+\sqrt[3]{3 x+5}} d x
$$

Perform the variable substitution:

$$
\begin{aligned}
& t=\sqrt[3]{3 x+5} \\
& x=\frac{t^{3}-5}{3} \\
& d x=t^{2} d t
\end{aligned}
$$

Recalculate the limits of integration:

$$
\begin{aligned}
& x_{1}=-\frac{5}{3} \Rightarrow t_{1}=\sqrt[3]{3 \cdot\left(-\frac{5}{3}\right)+5}=0 \\
& x_{2}=1 \Rightarrow t_{2}=\sqrt[3]{3 \cdot 1+5}=2
\end{aligned}
$$

Take the resulting integral:

$$
\begin{aligned}
& \int_{-5 / 3}^{1} \frac{\sqrt[3]{3 x+5}+2}{1+\sqrt[3]{3 x+5}} d x= \\
= & \int_{0}^{2} \frac{t^{2}(t+2)}{t+1} d t= \\
= & \int_{0}^{2}\left(t^{2}+t-1+\frac{1}{t+1}\right) d t= \\
= & \left.\left(\frac{t^{3}}{3}+\frac{t^{2}}{2}-t+\ln (t+1)\right)\right|_{0} ^{2}=\frac{8}{3}+2-2+\ln 3=\frac{8}{3}+\ln 3
\end{aligned}
$$

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%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_11-16%"

Categories: Kuznetsov's Problem Book Integrals Problem 11 | Integrals

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## Problem Kuznetsov Integrals 11-17

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