Let us consider the integral of the foll...

Let us consider the integral of the following forms
`f(x_(1), sqrt(mx^(2)+nx+p))^(1//2)`
Case I If `m gt 0`, then put `sqrt(mx^(2)+nx+C)=u pm x sqrt(m)`
Case II If `p gt 0`, then put `sqrt(mx^(2)+nx+C)=u x pm sqrt(p)`
Case III If quadratic equation `mx^(2)+nx+p=0` has real roots `alpha` and `beta` there put `sqrt(mx^(2)+nx+p)=(x-alpha) u ` or `(x-beta)u`
To evaluate `int(dx)/((x-1)sqrt(-x^(2)+3x-2))` one of the most suitable substitution could be

`sqrt(9x^(2)+4x+6)=upm3x`

`sqrt(9x^(2)+4x+6)=3upmx`

`x=(1)/(t)`

`9x^(2)+4x+6=(1)/(t)`

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The correct Answer is:

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Let us consider the integral of the following forms f(x_(1), sqrt(mx^(2)+nx+p))^(1//2) Case I If m gt 0 , then put sqrt(mx^(2)+nx+C)=u pm x sqrt(m) Case II If p gt 0 , then put sqrt(mx^(2)+nx+C)=u x pm sqrt(p) Case III If quadratic equation mx^(2)+nx+p=0 has real roots alpha and beta there put sqrt(mx^(2)+nx+p)=(x-alpha) u or (x-beta)u int ((x+sqrt(1+x^(2)))^(15))/(sqrt(1+x^(2))) dx is equal to

`((x+sqrt(1+x^(2)))^(16))/(10)+C`

`(1)/(15(sqrt(1+x^(2)+x)))+C`

`(15)/((sqrt(1+x^(2))-x))+C`

`((x+sqrt(1+x^(2)))^(15))/(15)+C`

`-a`

`+- m^(2)`

`n^(2)- 4m - 1 = 0 `

`n^(2) + 4m - 1 = 0 `

`m^(2) - 4n -1 =0`

`m^(2) - 4n - 1 = 0`