Advertisements
Advertisements
प्रश्न
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Advertisements
उत्तर
f '(x) = `"x"^2/2 - "kx" + 1` ...[Given]
f(x) = ∫ f '(x) dx
`= int ("x"^2/2 - "kx" + 1)`dx
`= 1/2 int "x"^2 "dx" - "k" int "x" "dx" + int 1 * "dx"`
`= 1/2 * "x"^3/3 - "k" ("x"^2/2) + "x" + "c"`
∴ f(x) = `"x"^3/6 - "k"/2 "x"^2 + "x" + "c"` ...(i)
Now, f(0) = 2
∴ `(0)^3/6 - "k"/2 (0)^2 + 0 + "c"` = 2
∴ c = 2 ...(ii)
Also f(3) = 5 ...[Given]
∴ `(3)^3/6 - "k"/2 (3)^2 + 3 + 2 = 5`
∴ `27/6 - "9k"/2 + 5 = 5`
∴ `9/2 - "9k"/2 = 0`
∴ `"9k"/2 = 9/2`
∴ k = 1 ....(iii)
Substituting (ii) and (iii) in (i), we get
f(x) = `"x"^3/6 - "x"^2/2 + "x" + 2`
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`e^(2x+3)`
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int "x" * "e"^"2x"` dx
Evaluate: `int "e"^sqrt"x"` dx
`int cot^2x "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
