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
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Find `intsqrtx/sqrt(a^3-x^3)dx`
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate `int 1/("x" ("x" - 1))` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int "x" * "e"^"2x"` dx
`int (cos x)/(1 - sin x) "dx" =` ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
`int x^3 e^(x^2) dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
