Advertisements
Advertisements
Question
`int(log(logx))/x "d"x`
Advertisements
Solution
Let I = `int(log(logx))/x "d"x`
Put log x = t
∴ `1/x "d"x` = dt
∴ I = `int log "t" "dt" = intlog"t"*1 "dt"`
= `log "t" int 1*"dt" - int ["d"/"dt"(log"t") int 1*"dt"]"dt"`
= `log "t"* "t" - int(1/"t" xx "t") "dt"`
= `"t"*log "t" - int "dt"`
= t log t − t + c
= t (log t − 1) + c
∴ I = logx [log (logx) − 1] + c
RELATED QUESTIONS
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
cot x log sin x
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Write a value of
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int "x" * "e"^"2x"` dx
`int (log x)/(log ex)^2` dx = _________
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int cos^3x dx` = ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
