English

Evaluate the following: ∫log⁡𝑥𝑥.𝑑⁢𝑥

Advertisements
Advertisements

Question

Evaluate the following: `int logx/x.dx`

Sum
Advertisements

Solution

Let I = `int logx/x.dx`

Put log x = t       

∴ `(1)/x.dx` = dt

∴ I = `int t.dt`

= `t^2/(2) + c`

= `(logx)^2/(2) + 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`


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


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 cos^(-1) x)/sqrt(1-x^2)`.


Integrate the function in `(xe^x)/(1+x)^2`.


Integrate the function in `((x- 3)e^x)/(x - 1)^3`.


Integrate the function in e2x sin x.


Evaluate the following:

`int x tan^-1 x . dx`


Evaluate the following:

`int sec^3x.dx`


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


Evaluate the following : `int x.cos^3x.dx`


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


Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`


Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`


Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.


Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`


Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`


Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)] 


Choose the correct options from the given alternatives :

`int tan(sin^-1 x)*dx` =


Choose the correct options from the given alternatives :

`int sin (log x)*dx` =


Choose the correct options from the given alternatives :

`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =


Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`


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


Integrate the following w.r.t.x : e2x sin x cos x


Evaluate the following.

`int x^2 e^4x`dx


Evaluate: Find the primitive of `1/(1 + "e"^"x")`


Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx


Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`


Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx


`int"e"^(4x - 3) "d"x` = ______ + c


`int (x^2 + x - 6)/((x - 2)(x - 1))  "d"x` = x + ______ + c


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


The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1)  dx` is


Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`


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


Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.


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


The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.


Evaluate `int tan^-1x  dx`


Complete the following activity:

`int_0^2 dx/(4 + x - x^2) `

= `int_0^2 dx/(-x^2 + square + square)`

= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`

= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`

= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`


Evaluate:

`inte^x "cosec"  x(1 - cot x)dx`


Evaluate the following.

`intx^3 e^(x^2)dx`


Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?


Evaluate \[\int e^x\sin x\,dx.\]


For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×