Advertisements
Advertisements
Question
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
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
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
The value of \[\int\frac{1}{x + x \log x} dx\] is
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : tan2x dx
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
`int 1/(cos x - sin x)` dx = _______________
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int logx/x "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int x^x (1 + logx) "d"x`
Choose the correct alternative:
`int(1 - x)^(-2) dx` = ______.
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int(5x + 2)/(3x - 4) dx` = ______
`int ("d"x)/(x(x^4 + 1))` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
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 ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))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`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate `int 1/(x(x-1))dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
