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antiseptic1488 [7]
3 years ago
14

Help guys soal integral​

Mathematics
1 answer:
Afina-wow [57]3 years ago
5 0

Answer:

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

Step-by-step explanation:

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

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