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Câu hỏi: Tìm nguyên hàm của hàm số $f(x)=\dfrac{1}{\sqrt{x+9}-\sqrt{x}}$.
A. $\dfrac{2}{3\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)}+C$.
B. $\dfrac{2}{27}\left(\sqrt{(x+9)^3}+\sqrt{x^3}\right)+C$.
C. $\dfrac{2}{27}\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)+C$.
D. $-\dfrac{2}{27}\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)+C$.
A. $\dfrac{2}{3\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)}+C$.
B. $\dfrac{2}{27}\left(\sqrt{(x+9)^3}+\sqrt{x^3}\right)+C$.
C. $\dfrac{2}{27}\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)+C$.
D. $-\dfrac{2}{27}\left(\sqrt{(x+9)^3}-\sqrt{x^3}\right)+C$.
Ta có $f(x)=\dfrac{1}{\sqrt{x+9}-\sqrt{x}}=\dfrac{\sqrt{x+9}+\sqrt{x}}{9}$
$
\int f(x) d x=\int \dfrac{\sqrt{x+9}+\sqrt{x}}{9} d x=\dfrac{1}{9} \int\left[(x+9)^{\dfrac{1}{2}}+x^{\dfrac{1}{2}}\right] d x=\dfrac{2}{27}\left(\sqrt{(x+9)^3}+\sqrt{x^3}\right)+C
$
$
\int f(x) d x=\int \dfrac{\sqrt{x+9}+\sqrt{x}}{9} d x=\dfrac{1}{9} \int\left[(x+9)^{\dfrac{1}{2}}+x^{\dfrac{1}{2}}\right] d x=\dfrac{2}{27}\left(\sqrt{(x+9)^3}+\sqrt{x^3}\right)+C
$
Đáp án B.
