Advertisements
Advertisements
Question
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Advertisements
Solution
Let I = `int(sin(logx)^2)/x.log.x.dx`
Put (logx)2 = t
∴ `2logx. 1/x.dx` = dt
∴ `1/xlogx.dx = (1)/(2)dt`
∴ I = `(1)/(2) int sin t.dt`
= `-(1)/(2) cost + c`
= `-(1)/(2)cos[(logx)^2] + 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`
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x sin 3x.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following: `int logx/x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
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`
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
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^4x`dx
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("9x"^2 - 25)`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
∫ log x · (log x + 2) dx = ?
`int log x * [log ("e"x)]^-2` dx = ?
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int1/sqrt(x^2 - a^2) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
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
Evaluate:
`inte^x sinx dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
