1.

int(dx)/((x^(2)-4)sqrt(x))=-1/(2sqrt2) f(x) + 1/(4sqrt2) log|(sqrt(x) - sqrt(2))/(sqrt(x) + sqrt(2))| +c where c is constant of intergration and f(2) = pi/4 and f(6) + f(2/3) is equal to

Answer»

`pi/4`
`pi/2`
`(3 pi)/4`
`(pi)/(-4)`

Solution :`l = f(DX)/((x^(2) - 4)sqrt(x))`
`x = l^(2) :. Dx = 2tdt`
`l = int (2tdt)/((t^4 - 4)t) = 2 int (dt)/((t^(2) + 2)(t^(2) - 2))`
`-1/2 int (dt)/(t^2 + 2) + 1/2 int (dt)/(t^2 - 2)`
`(-1)/(2SQRT2) TAN^(-1) (t/(sqrt2)) + 1/(4sqrt2) log |(t-sqrt2)/(t + sqrt2)| + C`
`=(-1)/(2sqrt2) tan^(-1) sqrt(x/2) + 1/(4sqrt2) log |(sqrt(x) - sqrt(2))/(sqrt(x) + sqrt(2))|+C`
`f(x) = tan^(-1) ((SQRTX)/(sqrt2))`.


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