Advertisements
Advertisements
प्रश्न
Solve:
dy/dx = cos(x + y)
Advertisements
उत्तर
Given,
`dy/dx= cos (x + y)` …(i)
Put `x + y = v` …(ii)
`∴ y = v – x`
`∴ dy/dx=(dv)/dx-1` …(iii)
Substituting (ii) and (iii) in (i), we get
`(dv)/dx-1=cosv`
`therefore (dv)/dx=1+cosv`
`therefore (dv)/dx=2cos^2(v/2)`
`therefore 1/cos^2(v/2)dv=2dx`
`therefore sec^2(v/2)dv=2dx`
Integrating on both sides, we get
`int sec^2(v/2)dv=2intdx`
`therefore 2tan(v/2)=2x+c'`
`therefore tan(v/2)=x+(c')/2`
`therefore tan((x+y)/2)=x+c`, where `c=(c')/2`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals:
tan2x dx
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
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(3x^2 - 8).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`
Evaluate the following : `int (logx)2.dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
Evaluate: `int "x" * "e"^"2x"` dx
`int (log x)/(log ex)^2` dx = _________
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int 1/("x"("x" - 1)) "dx"`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate:
`int sqrt((a - x)/x) dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
Which standard substitution is used for \[\sqrt{x^2+\mathrm{a}^2}\], \[\frac{1}{\sqrt{x^2+\mathrm{a}^2}}\], or \[x^2+\mathrm{a}^2\]?
After choosing \[u=g(x)\], what is the next step?
