Advertisements
Advertisements
प्रश्न
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in tan-1 x.
Integrate the function in x (log x)2.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x tan^-1 x .dx`
Evaluate the following:
`int sec^3x.dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int"e"^(4x - 3) "d"x` = ______ + c
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int cot "x".log [log (sin "x")] "dx"` = ____________.
Find `int_0^1 x(tan^-1x) "d"x`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate the following.
`intx^3 e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
