Advertisements
Advertisements
Question
Find :
`∫(log x)^2 dx`
Advertisements
Solution
`∫(log x)^2 dx`
let `u = (logx)^2 , "v" = 1`
`∫u."v" dx = u∫"v"dx - ∫[(du)/dx∫"v"dx]dx`
`therefore ∫ (log x)^2 . 1dx = (log x)^2 ∫1dx - ∫[2log x xx 1/x xx xdx]`
= `x(log|x|^2) - 2∫log x dx`
`x(log x)^2 - 2(x log|x| - x) + C`
= `x(log|x|)^2 - 2x log|x| + 2x + C` .
APPEARS IN
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
Integrate the function in x log 2x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
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 [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int(logx)^2dx` equals ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
Repeated parts may be needed for which pair of integrals?
