Advertisements
Advertisements
Question
Evaluate the following : `int log(logx)/x.dx`
Advertisements
Solution
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
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in `x^2e^x`.
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in x tan-1 x.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
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` =
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int(x + 1/x)^3 dx` = ______.
`int logx/(1 + logx)^2 "d"x`
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int(logx)^2dx` equals ______.
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((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`int(1-x)^-2 dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`inte^(xloga).e^x dx` is ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate `int tan^-1x dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
Which function is an example under priority \(I\) in the LIATE rule?
Using \[t=1-x^2,\] what is \[\int \frac{x\,dx}{\sqrt{1-x^2}}?\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
