मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

Integrate the following w.r.t.x : log (log x)+(log x)–2 - Mathematics and Statistics

Advertisements
Advertisements

प्रश्न

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

बेरीज
Advertisements

उत्तर

Let I = `int [log (logx) + (logx)^-2]*dx`

= `int [log(logx) + 1/(log x)^2]*dx`
Put log x = t
∴ x = et
∴ x = et·dt

∴ I = `int (log t + 1/t^2)e^t*dt`

= `int e^t (log t + 1/t - 1/t + 1/t^2)*dt`

= `int [e^t (log t 1/t) + e^t (-1/t + 1/t^2)]*dt`

= `inte^t (log t + 1/t)*dt - int e^t (1/t - 1/t^2)*dt`

= I1 – I2 

In I1, Put f(t) = log t. Then f'(t) = `(1/t)`

∴ I1 = `int e^t [f(t) + f'(t)]*dt`

= `e^t f(t)`

= `e^t log t`

In I2, Put g(t) = `(1/t)`. Then g'(t) = `-(1/t^2)`

∴ I2 = `int e^t ["g"(t) + "g"'(t)]*dt`

= `e^t "g" (t)`

= `e^t*(1/t)`

∴ I = `e^t log t - e^t/t + c`

= `xlog (logx) - x/logx + c`.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 3: Indefinite Integration - Miscellaneous Exercise 3 [पृष्ठ १५०]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
पाठ 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 3.08 | पृष्ठ १५०

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Integrate the function in x cos-1 x.


Integrate the function in (sin-1x)2.


Evaluate the following : `int x^3.tan^-1x.dx`


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


Integrate the following functions w.r.t. x : `e^(2x).sin3x`


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

`e^-x cos2x`


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

sin (log x)


Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`


Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`


Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]e 


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


Choose the correct options from the given alternatives :

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


Choose the correct options from the given alternatives :

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


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


Integrate the following w.r.t.x : log (x2 + 1)


Solve the following differential equation.

(x2 − yx2 ) dy + (y2 + xy2) dx = 0


Choose the correct alternative from the following.

`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5  "dx"` = 


Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx


Evaluate: `int "dx"/("9x"^2 - 25)`


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


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


`int (sin(x - "a"))/(cos (x + "b"))  "d"x`


`int 1/sqrt(2x^2 - 5)  "d"x`


`int ("e"^xlog(sin"e"^x))/(tan"e"^x)  "d"x`


Choose the correct alternative:

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


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


`int "e"^x x/(x + 1)^2  "d"x`


`int cot "x".log [log (sin "x")] "dx"` = ____________.


`int log x * [log ("e"x)]^-2` dx = ?


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 = ______.


Find: `int e^x.sin2xdx`


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


Solve: `int sqrt(4x^2 + 5)dx`


`int(logx)^2dx` equals ______.


`int_0^1 x tan^-1 x  dx` = ______.


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


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


Evaluate the following.

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


`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`


Evaluate `int(3x-2)/((x+1)^2(x+3))  dx`


`inte^(xloga).e^x dx` is ______


`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`


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 the following.

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


Evaluate the following.

`intx^3/sqrt(1+x^4)  dx`


If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.


Evaluate:

`inte^x "cosec"  x(1 - cot x)dx`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×