English

Evaluate the following : ∫x2.logx.dx

Advertisements
Advertisements

Question

Evaluate the following : `int x^2.log x.dx`

Sum
Advertisements

Solution

Let I = `int x^2.logx.dx`

= `int log x.x^2.dx`

= `(logx) int x^2.dx - int[{d/dx (logx) int x^2.dx}].dx`

= `(logx).x^3/(3) - int (1)/x.x^3/(3).dx`

= `x^3/(3) logx - (1)/(3) int x^2.dx`

= `x^3/(3) logx - (1)/(3)(x^3/3) + c`

= `x^3/(9)(3.logx - 1) + c`.

shaalaa.com
  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.3 [Page 137]

APPEARS IN

Video TutorialsVIEW ALL [1]

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 xlog x.


Integrate the function in x sin−1 x.


Integrate the function in x cos-1 x.


Integrate the function in tan-1 x.


Integrate the function in `e^x (1/x - 1/x^2)`.


`int e^x sec x (1 +   tan x) dx` equals:


Evaluate the following : `int x^2tan^-1x.dx`


Evaluate the following : `int log(logx)/x.dx`


Evaluate the following : `int cos(root(3)(x)).dx`


Integrate the following functions w.r.t. x:

sin (log x)


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` =


Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`


Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`


Integrate the following w.r.t.x : log (log x)+(log x)–2 


Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`


Evaluate the following.

∫ x log x dx


Evaluate the following.

`int "e"^"x" "x"/("x + 1")^2` dx


Evaluate the following.

`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx


Evaluate the following.

`int (log "x")/(1 + log "x")^2` dx


`int ("x" + 1/"x")^3 "dx"` = ______


Choose the correct alternative from the following.

`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` = 


Evaluate: `int "dx"/(5 - 16"x"^2)`


Choose the correct alternative:

`intx^(2)3^(x^3) "d"x` =


Evaluate `int 1/(x log x)  "d"x`


`int 1/sqrt(x^2 - 8x - 20)  "d"x`


The value of `int "e"^(5x) (1/x - 1/(5x^2))  "d"x` is ______.


`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.


Evaluate the following:

`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`


`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


The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x))  dx` is


If u and v ore differentiable functions of x. then prove that:

`int uv  dx = u intv  dx - int [(du)/(d) intv  dx]dx`

Hence evaluate `intlog x  dx`


`int 1/sqrt(x^2 - 9) dx` = ______.


Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`


Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.


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` = ______.


`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^x ((1 - sinx)/(1 - cosx))dx`.


Evaluate:

`intcos^-1(sqrt(x))dx`


Evaluate the following.

`intx^3/sqrt(1+x^4)dx`


Evaluate:

`int x^2 cos x  dx`


Evaluate `int(1 + x + x^2/(2!))dx`.


The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×