Question

Integrate the following functions with respect to x :

`1/(sqrt(x + 3) - sqrt(x - 4))`

Answer 1

`int 1/(sqrt(x + 3) - sqrt(x - 4))  "d"x`

= `int 1/(sqrt(x + 3) - sqrt(x - 4)) xx (sqrt(x + 3) + sqrt(x - 4))/(sqrt(x + 3) + sqrt(x - 4))  "d"x`

= `int (sqrt(x + 3) + sqrt(x - 4))/((x + 3) - (x - 4)) "d"x`

= `int (sqrt(x + 3) + sqrt(x - 4))/(x + 3 - x + 4)  "d"x`

= `int (sqrt(x + 3) + sqrt(x - 4))/7 "d"x`

= `int sqrt(x + 3)/7  "d"x + int sqrt(x - 4)/7  "d"x`

= `1/7 int (x + 3)^(1/2)  "d"x + 1/7 int (x - 4)^(1/2)  "d"x`

= `1/7 (x + 3)^(1/2 + 1)/(1/2 + 1) + 1/7 (x - 4)^(1/2 + 1)/(1/2 + 1) + "c"`

= `1/7 (x + 3)^(3/2)/(3/2) + 1/7 (x - 4)^(3/2)/(3/2) + "c"`

= `1/7 xx 2/3 (x + 3)^(3/2) + 1/7 xx 2/3 (x - 4)^(3/2) + "c"`

= `2/21 [(x + 3)^(3/2) + (x - 4)^(3/2)] + "c"`