Advertisements
Advertisements
Question
Integrate the function in x log x.
Advertisements
Solution
Let `I = int x log x dx`
`= log x int x dx - int [d/dx (log x) int x dx] dx`
`= log x (x^2/2) - int (1/x * x^2/2) dx`
`= x^2/2 log x - 1/2 int x dx + C`
`= x^2/2 log x -1/2 xx x^2/2 + C`
`= x^2/2 log x - 1/4 x^2 + C`
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) dx` equals:
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x.sin^2x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate:
∫ (log x)2 dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^2) dx` = ______
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
