Tìm nguyên hàm của hàm số sau: ∫dx/x+1√+x√

Question

\(\int \frac{{\rm d}x}{\sqrt{x\, +\, 1} \, \, +\, \sqrt{x} } \, \)

Answer 1

\(\int \frac{{\rm d}x}{\sqrt{x\, +\, 1} \, +\, \sqrt{x} } \,  =\, \int \left(\sqrt{x\, +\, 1} \, -\, \sqrt{x} \right){\rm d}x =\int \sqrt{x+1} {\rm d}x -\int \sqrt{x} {\rm d}x =\int \left(x+1\right)^{\frac{1}{2} } {\rm d}\left(x+1\right)\,  -\int x^{\frac{1}{2} } {\rm d}x \)
\(=\, \frac{2}{3} \, \left(x\, +\, 1\right)^{\frac{3}{2} } \, -\, \frac{2}{3} \, x^{\frac{3}{2} } \, +\, C\, =\, \frac{2}{3} \, \left(x\, +\, 1\right)\, \sqrt{x\, +\, 1} \, -\, \frac{2}{3} \, x\, \sqrt{x} \, +\, C\)