Biết $\Large \int_{1}^{3} \frac{ d x}{\sqrt{x+1}-\sqrt{x}}=a \sqrt{3}+

Biết $\Large \int_{1}^{3} \frac{ d x}{\sqrt{x+1}-\sqrt{x}}=a \sqrt{3}+b \sqrt{2}+c$ với a, b, c là các số hữu tỷ. Tính $\Large P=a+b+c$

• A. $\Large P=5$
• B. $\Large P=\dfrac{2}{3}$
• C. $\Large P=\dfrac{13}{2}$
• D. $\Large P=\dfrac{16}{3}$

Ta có: $\Large \int_{1}^{3} \dfrac{ d x}{\sqrt{x+1}-\sqrt{x}}=\int_{1}^{3}(\sqrt{x+1}+\sqrt{x}) d x$$\Large =\left.\left[\dfrac{2}{3} \sqrt{(x+1)^{3}}+\dfrac{2}{3} \sqrt{x^{3}}\right]\right|_{1} ^{3}=2 \sqrt{3}-\dfrac{4}{3} \sqrt{2}+\dfrac{14}{3}$

Khi đó $\Large \left\{\begin{array}{l}
a=2 \\
b=-\dfrac{4}{3} \Rightarrow P=a+b+c=\dfrac{16}{3} . \\
c=\dfrac{14}{3}
\end{array}\right.$