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
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x sin 3x.
Integrate the function in x log x.
Integrate the function in x cos-1 x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.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 : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*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 tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int x^2 e^4x`dx
`int 1/(4x + 5x^(-11)) "d"x`
`int ("d"x)/(x - x^2)` = ______
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
∫ log x · (log x + 2) dx = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) 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
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`inte^x sinx dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
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:
`int (sin(x - a))/(sin(x + a))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)
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
Which pair is listed as Trigonometric in the LIATE rule?
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
