Advertisements
Advertisements
Question
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Advertisements
Solution
By the definition of integral,
f(x) = `int f'(x).dx`
= `int (x - 3/x^3).dx`
= `int x dx - 3 int x^-3 .dx`
= `x^2/(2) - (3x^((-2)))/((-2)) + c`
= `x^2/(2) + (3)/(2x^2) + c` ...(1)
f(1) = `(11)/(2)` ...(Given)
∴ `(1)/(2) + (3)/(2) + c = (11)/(2)`
∴ c = `(7)/(2)`
∴ f(x) = `x^2/(2) + (3)/(2x^2) + (7)/(2)` ...[By (1)]
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`(1+ log x)^2/x`
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
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
Evaluate: `int log ("x"^2 + "x")` dx
`int sqrt(1 + sin2x) dx`
`int logx/x "d"x`
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int ("d"x)/(x(x^4 + 1))` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate:
`int sin^2(x/2)dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1+x^4)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).
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
