Advertisements
Advertisements
प्रश्न
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Advertisements
उत्तर
f'(x) = x2 + 5 ...(Given)
∴ f(x) = ∫f'(x) dx
∴ f(x) = ∫(x2 + 5) dx
∴ f(x) = ∫ x2 dx + 5 ∫ dx
∴ f(x) = `"x"^3/3 + 5"x" + "c"` ....(i)
Substitute x = 0, f(0) = −1 ...(Given)
∴ f(x) = `"x"^3/3 + 5"x" + "c"`
∴ f(0) = `0^3/3 + 5(0) + "c"`
∴ −1 = 0 + 0 + c
∴ c = −1
Substituting c = – 1 in (i), we get,
∴ f(x) = `"x"^3/3 + 5"x" + (− 1)`
∴ f(x) = `"x"^3/3 + 5"x" − 1`
APPEARS IN
संबंधित प्रश्न
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Write a value of
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int ("d"x)/(x(x^4 + 1))` = ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
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 ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
