Question

\int x \sqrt { x + 3 d x }

Answer 1

$\int\limits {x\sqrt{x+3} } \, dx$

u = x + 3   --|> (x = u - 3)

du = dx

$\int\limits {(u-3)\sqrt{u} } \, du =\int\limits {(u-3)(u^{\frac{1}{2} }) } \, du \\\\\int\limits {u^{\frac{3}{2} }-3u^{\frac{1}{2} } } \, du \\\\\frac{u^{\frac{5}{2} } }{\frac{5}{2} } -\frac{3u^{\frac{3}{2} } }{\frac{3}{2} } \\\\\frac{2u^{\frac{5}{2} } }{5} -\frac{6u^{\frac{3}{2} } }{3} \\\\\frac{2(x+3)^{\frac{5}{2} } }{5} -\frac{6(x+3)^{\frac{3}{2} } }{3} \\\\\frac{2\sqrt{(x+3)^5} }{5} -{2\sqrt{(x+3)^3} }+C$