Question

Plz solve the calculus :')

Answer 1

`(2)/(3) \bigg[ x\sqrt{ {x}} + (x - 1)\sqrt{ {(x - 1)}} \bigg] + c`

Explanation:

`\int (1)/( √(x) - √(x - 1) ) dx \\ \\ = \int ( √(x) + √(x - 1))/( (√(x) - √(x - 1) )( √(x) + √(x - 1) )) dx \\ \\ = \int ( √(x) + √(x - 1))/( (√(x))^(2) - (√(x - 1) )^(2)) dx \\ \\ = \int ( √(x) + √(x - 1))/( x - (x - 1 )) dx \\ \\ = \int ( √(x) + √(x - 1))/( x - x + 1) dx \\ \\ = \int ( √(x) + √(x - 1))/( 1) dx \\ \\ = \int (√(x) + √(x - 1)) dx \\ \\ = \int √(x) \: dx+ \int√(x - 1) \: dx \\ \\ = \int {x}^{ (1)/(2) } \: dx+ \int {(x - 1)}^{ (1)/(2) } \: dx \\ \\ = \frac{ {x}^{ (3)/(2) } }{ (3)/(2) } + \frac{ {(x - 1)}^{ (3)/(2) } }{ (3)/(2) } + c \\ \\ = (2)/(3) {x}^{ (3)/(2) } + (2)/(3) {(x - 1)}^{ (3)/(2) } + c \\ \\ = (2)/(3) \bigg[ \sqrt{ {x}^(3) } + \sqrt{ {(x - 1)}^(3) } \bigg] + c \\ \\ = \bold{\purple {(2)/(3) \bigg[ x\sqrt{ {x}} + (x - 1)\sqrt{ {(x - 1)}} \bigg] + c}}`