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(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + 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
`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 x sin 3x.
Integrate the function in x log 2x.
Integrate the function in x2 log x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in (x2 + 1) log x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
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`
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int (sinx)/(1 + sin x) "d"x`
`int sin4x cos3x "d"x`
`int ("d"x)/(x - x^2)` = ______
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int e^(logcosx)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`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
