Advertisements
Advertisements
Question
Integrate the function in x (log x)2.
Solution
Let `I = int x (log x)^2 dx`
`= int (log x)^2 * x dx`
`= (log x)^2 int x dx - int [d/dx (log x)^2 * int x dx] dx`
`= x^2/2 (log x)^2 - int (log x) * x dx + C`
`= x^2/2 (log x)^2 - [ (log x) * x^2/2 - int 1/x * x^2/2 dx]`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int x dx`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int*x^2/2 + C`
`= x^2 (log x)^2 - x^2/2 log x + 1/2 * x^2/2 + C`
APPEARS IN
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^2 "e"^"3x"`dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x(x - 1)) "d"x`
∫ log x · (log x + 2) dx = ?
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Find: `int e^x.sin2xdx`
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
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`.
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
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^3e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
