## Problem Statement

Calculate the definite integral:

$$
\int_{1}^{2} \frac{x+\sqrt{3 x-2}-10}{\sqrt{3 x-2}+7} d x
$$
## Solution

11.10. $\int_{1}^{2} \frac{x+\sqrt{3 x-2}-10}{\sqrt{3 x-2}+7} d x=$

saucera: $\sqrt{3 x-2}=t \Rightarrow x=\frac{1}{3}\left(t^{2}+2\right)$, $d x=\frac{2}{3} t d t$

$=\frac{2}{3} \int_{1}^{2} \frac{\frac{1}{3}\left(t^{2}+2\right)+t-10}{t+7} t d t=\frac{2}{9} \int_{1}^{2} \frac{t^{3}+3 t^{2}-28 t}{t+7} d t=$

$=\frac{2}{9} \int_{1}^{2}\left(t^{2}-4 t\right) d t=\left.\frac{2}{9}\left(\frac{1}{3} t^{3}-2 t^{2}\right)\right|_{1} ^{2}=$

$=\frac{2}{9}\left(\frac{8}{3}-8-\frac{1}{3}+2\right)=-\frac{22}{27}$

![](https://cdn.mathpix.com/cropped/2024_05_22_cd7234dbad9f370b4719g-15.jpg?height=100&width=98&top_left_y=1445&top_left_x=128)

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Categories: In the form of a scan | Kuznetsov's Problem Book Integrals Problem 11 | Integrals

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## Problem Kuznetsov Integrals 11-11

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