Advertisements
Advertisements
प्रश्न
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Advertisements
उत्तर
f'(x) = 4x3 − 3x2 + 2x + k ....[Given]
f(x) = ∫ f'(x) dx
= ∫ (4x3 − 3x2 + 2x + k) dx
= 4 ∫ x3 dx − 3 ∫ x2 dx + 2 ∫ x dx + k ∫ dx
= `4 ("x"^4/4) - 3("x"^3/3) + 2("x"^2/2) "kx" + "c"`
∴ f(x) = x4 − x3 + x2 + kx + c ....(i)
Now, f(0) = 1 ...[Given]
∴ (0)4 − (0)3 + (0)2 + k(0) + c = 1
∴ c = 1 ....(ii)
Also, f(1) = 4 ....[Given]
∴ 1 − 1 + 1 + k + c = 4
∴ 1 + k + 1 = 4
∴ 2 + k = 4
∴ k = 2 ...(iii)
Substituting (ii) and (iii) in (i), we get
f(x) = x4 − x3 + x2 + 2x + 1
APPEARS IN
संबंधित प्रश्न
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
The value of \[\int\frac{1}{x + x \log x} dx\] is
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t.x:
cos8xcotx
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int x^3"e"^(x^2) "d"x`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int 1/(x(x-1)) dx`
