Question

integrate the function x√(x + 2).dx

Answer 1

$\bf{I=\int{x\sqrt{(x+2)}}\,dx}$

Let (x + 2) = t => x = (t - 2) -----(1)

differentiate both sides,

dx = dt -------(2)

put equations (1) and (2),

I = $\bf{\int{(t-2)\sqrt{t}}\,dt}$

= $\bf{\int{t\sqrt{t}}\,dt-2\int{\sqrt{t}}\,dt}$

= $\bf{\int{t^{3/2}}\,dt-2\int{t^{1/2}}\,dt}$

= $\bf{\frac{t^{3/2+1}}{3/2+1}-2\frac{t^{1/2+1}}{1/2+1}+C}$

= $\bf{\frac{2}{5}t^{5/2}-\frac{4}{3}t^{3/2}+C}$

put t = (x + 2)

I = $\bf{\frac{2}{5}(x+2)^{5/2}-\frac{4}{3}(x+2)^{3/2}+C}$