Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Advertisements
Solution
Let I = `int e^x .(1/x - 1/x^2).dx`
Let f(x) = `(1)/x`
∴ f'(x) = `-(1)/x^2`
∴ I = `int e^x[f(x) + f'(x)].dx`
= ex f(x) + c
= `e^x . (1)/x + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
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^2.log x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
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 (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int sin4x cos3x "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/sqrt(x^2 - a^2) dx` = ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
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 ______.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
