Evaluate: `intx/(sqrt(x+4))\ dx` (ii) `int(2-3x)/(sqrt(1+3x))\ dx`

Explanation

(i)
`int frac{x}{sqrt{x+4}} d x `

`= int(frac{x+4-4}{sqrt{x+4}}) d x `

`= int(sqrt{x+4}-frac{4}{sqrt{x+4}}) d x `

`=.int(x+4)^{frac{1}{2}} d x-4](x+4)^{-frac{1}{2}} d x `

`= frac{(x+4)^{frac{1}{2}}+1}{frac{1}{2}+1}-4 frac{[x+4]^{-frac{1}{2}+1}}{-frac{1}{2}+1}+C `

`= frac{2}{3}(x+4)^{frac{3}{2}}-8(x+4)^{frac{1}{2}}+C `

`=(x+4)^{frac{1}{2}}[frac{2}{3}(x+4)-8]+C `

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