Advertisements
Advertisements
Question
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Let 4ex – 25 = `A(2e^x - 5) + B d/(dx) (2e^x - 5)`
= 2exA – 5A + B(2ex)
2 × 2ex – 25 = 2ex (A + B) – 5A
∴ A + B = 2
And –25 = –5A
∴ A = 5
5 + B = 2
∴ B = 2 – 5 = – 3.
APPEARS IN
RELATED QUESTIONS
`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 (sin-1x)2.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int x^3.logx.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int ("d"x)/(x - x^2)` = ______
`int(x + 1/x)^3 dx` = ______.
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int_0^pi x log sin x "d"x`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
