Question

intigrate x/(a+bx)^1/2

Answer 1

integral of x is x^2÷2
integral of 1/(a+bx)^1/2 is{ (a+bx)^-1/2+1}÷-1/2+1 ×b
= (a+bx)^1/2÷a×1/2
total value is
x^2×a×1/2
_________( division line)
2×(a+bx)^1/2

Answer 2

Hi,

$t²=a+bx\\x=\dfrac{t^2-a}{b} \\dx=\dfrac{2tdt}{b} \\\int{\dfrac{x}{\sqrt{a+bt}}}} \, dx =\int { \dfrac{ (t^2-a)*2t }{bt*b}} \, dx \\=\dfrac{2}{b^2}\int {( t^2-a)} \, dt \\ =\dfrac{2}{b^2}(\dfrac{t^3}{3}-at)+C\\=\dfrac{2}{b^2}(\dfrac{\sqrt{a+bx}}{3}-a\sqrt{a+bx})+C$