Advertisements
Advertisements
Question
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Advertisements
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`
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int \log_e x\ dx\].
Write a value of
The value of \[\int\frac{1}{x + x \log x} dx\] is
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following.
`intxsqrt(1+x^2)dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
