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Question
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
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Solution
f '(x) = `sqrt"x"` ....[Given]
f(x) = ∫ f '(x)
`= int sqrt"x"` dx
`= int "x"^(1/2)` dx
`= "x"^(3/2)/(3/2)` + c
∴ f(x) = `2/3 "x"^(3/2) + "c"` ...(i)
Now, f(1) = 2 ....[Given]
∴ `2/3 (1)^(3/2) + "c" = 2`
∴ c = `2 - 2/3`
∴ c = `4/3`
∴ f(x) = `2/3 "x"^(3/2) + 4/3`
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