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Question
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Options
`(1)/(2)sqrt(x + 1) + c`
`(2)/(3)(x + 1)^(3/2) + c`
`sqrt(x + 1) + c`
`2(x - 1)^(3/2) + c`
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Solution
`(2)/(3)(x + 1)^(3/2) + c`
Explanation:
`I = int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx`
I = `int sqrt((1+x)^2+ sqrtx *sqrt(1+ x) )/ (sqrt(x) + sqrt(1+x))*dx`
I = `int( sqrt(1 + x) sqrt(1 + x) + sqrtx )/ (sqrt(x)+sqrt(1 + x))*dx`
I `= int(sqrt(1+x)) dx = 2/3 (x + 1)^(3/2) + c`
I = `(2)/(3)(x + 1)^(3/2) + c`
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