Campos Brooks
11/13/2023 · Junior High School
\( \int _{}^{}\frac{1}{\sqrt {x}+2} d x \)
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Step-by-step Solution
It seems that the integral calculation is not supported directly. Let's try to simplify the integral expression first.
Given integral: \( \int \frac{1}{\sqrt{x}+2} dx \)
To simplify this integral, we can use the substitution method. Let \( u = \sqrt{x} \), then \( du = \frac{1}{2\sqrt{x}} dx \) and \( dx = 2u du \).
Substitute \( u = \sqrt{x} \) and \( dx = 2u du \) into the integral expression:
\( \int \frac{1}{\sqrt{x}+2} dx = \int \frac{1}{u+2} 2u du \)
Now, we can simplify the integral expression and calculate the result.
Evaluate the integral by following steps:
- step0: Evaluate:
\(\int \frac{1}{u+2}\times 2u du\)
- step1: Multiply the terms:
\(\int \frac{2u}{u+2} du\)
- step2: Use properties of integrals:
\(2\times \int \frac{u}{u+2} du\)
- step3: Rewrite the fraction:
\(2\times \int 1-\frac{2}{u+2} du\)
- step4: Use properties of integrals:
\(2\left(\int 1 du-\int \frac{2}{u+2} du\right)\)
- step5: Evaluate the integral:
\(2\left(u-2\ln{\left(\left|u+2\right|\right)}\right)\)
- step6: Evaluate the integral:
\(2u-4\ln{\left(\left|u+2\right|\right)}\)
- step7: Add the constant of integral C:
\(2u-4\ln{\left(\left|u+2\right|\right)} + C, C \in \mathbb{R}\)
The simplified result of the integral \( \int \frac{1}{\sqrt{x}+2} dx \) is:
\[ 2u - 4\ln{|u+2|} + C \]
where \( u = \sqrt{x} \) and \( C \) is the constant of integration.
Quick Answer
Решите интеграл \( \int \frac{1}{\sqrt{x}+2} dx \) с помощью подстановки \( u = \sqrt{x} \). Результат: \( 2u - 4\ln{|u+2|} + C \), где \( u = \sqrt{x} \) и \( C \) — константа интегрирования.
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