Question

Help guys soal integral​

Answer 1

E.
`\purple { \bold{ (2)/(3) ( (x - 1)/(x) ) \sqrt{(x - 1)/(x) } + c }}`

Explanation:

`\int \sqrt{ \frac{x - 1}{ {x}^(5) } } dx \\ \\ = \int \sqrt{ \frac{x - 1}{ {x}^(4) .x} } dx \\ \\ = \int \frac{1}{ {x}^(2)}\sqrt{ (x - 1)/( x) } dx \\ \\ = \int \frac{1}{ {x}^(2)}\sqrt{ 1 - (1)/( x) } dx \\ \\ let \: 1 - (1)/( x) = t \\ \\ \implies \: \frac{1}{ {x}^(2) } dx = dt \\ \\ \implies \int \frac{1}{ {x}^(2)}\sqrt{ 1 - (1)/( x) } dx = \int √(t) dt \\ \\ = \int {t}^{ (1)/(2) } dt \\ \\ = \frac{t ^{ (3)/(2) } }{ (3)/(2) } + c \\ \\ = (2)/(3) t ^{ (3)/(2) } + c \\ \\ = (2)/(3) \sqrt{ {t}^(3) } + c \\ \\ = (2)/(3) t\sqrt{ {t} } + c \\ \\ = (2)/(3) (1 - (1)/(x) ) \sqrt{1 - (1)/(x) } + c \\ \\ \red{ \bold{= (2)/(3) ( (x - 1)/(x) ) \sqrt{(x - 1)/(x) } + c }}\\ \\`