Advertisements
Advertisements
प्रश्न
Find :
`∫(log x)^2 dx`
उत्तर
`∫(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
संबंधित प्रश्न
Integrate the function in x sin 3x.
Integrate the function in x log x.
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
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_0^1 x tan^-1 x dx` = ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
