Advertisements
Advertisements
प्रश्न
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Advertisements
उत्तर
Given differential equation is `e^((dy)/(dx)) = x^2`
Taking log both sides, we get
`(dy)/(dx) loge` = 2 logx
⇒ `(dy)/(dx)` = 2 logx ...[∵ loge = 1]
⇒ dy = 2 logx dx
On integrating both sides, we get
`intdy = 2intlogxdx`
⇒ y = `2int1.logxdx`
⇒ y = `[logx int1dx - int d/(dx) (logx)(int1.dx)dx]`
⇒ y = `2[logx(x) - int1/x (x)dx]` ...[Using integration by parts]
⇒ y = 2[xlogx – x] + C
⇒ y = 2x(logx – 1) + C
संबंधित प्रश्न
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x log 2x.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x log x) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate the following:
`intx^3e^(x^2)dx`
The value of `inta^x.e^x dx` equals
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
Integration by parts is a method of integration based on which rule of differentiation?
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
