Advertisements
Advertisements
प्रश्न
Evaluate the following : `int log(logx)/x.dx`
Advertisements
उत्तर
Let I = `int log(logx)/x.dx`
= `int log(logx). 1/xdx`
Put log = t
∴ `1/x.dx = dt`
∴ I = `int logt dt`
= `int (logt).1dt`
= `(logt) int 1dt - int[d/d (logt) int 1dt]dt`
= `(log t)t - int 1/t xx tdt`
= `t log t - int 1dt`
= t logt t – t + c
= t(log t – 1) + c
= (log x).[log(log x) – 1] + c.
APPEARS IN
संबंधित प्रश्न
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin x.
Integrate the function in x log x.
Integrate the function in x cos-1 x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate `int 1/(x(x - 1)) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int 1/sqrt(x^2 - a^2)dx` = ______.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
`int_0^1 x tan^-1 x dx` = ______.
`int1/sqrt(x^2 - a^2) dx` = ______
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
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(logcosx)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
