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

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

Advertisements
Advertisements

प्रश्न

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

बेरीज
Advertisements

उत्तर

Let I = `int log (1 + x)^((1 + x)).dx`

= `int (1 + x)log(1 + x).dx`

= `int [log(1 + x)] (1 + x).dx`

= `[log(1 + x) int (1 + x).dx - int[d/dt {log(1 + x)} int (1 + x).dx].dx`

= `[log (1 + x)] [(1 + x)^2/2] - int 1/(x + 1).(x + 1)^2/(2).dx`

= `(x + 1)^2/(2).log(1 + x) - (1)/(2) int (x + 1).dx`

= `(x + 1)^2/(2).log (1 + x) - (1)/(2).(x + 1)^2/(2) + c`

= `(x + 1)^2/(2)[log (1 + x) - 1/2] + c`.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 3: Indefinite Integration - Exercise 3.3 [पृष्ठ १३८]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
पाठ 3 Indefinite Integration
Exercise 3.3 | Q 3.8 | पृष्ठ १३८

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Integrate : sec3 x w. r. t. x.


Integrate the function in xlog x.


Integrate the function in x cos-1 x.


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


Integrate the function in x (log x)2.


Integrate the function in (x2 + 1) log x.


Integrate the function in ex (sinx + cosx).


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


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


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


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

`e^-x cos2x`


Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`


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


If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.


Choose the correct options from the given alternatives :

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


Choose the correct options from the given alternatives :

`int [sin (log x) + cos (log x)]*dx` =


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


Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`


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


Integrate the following with respect to the respective variable : cos 3x cos 2x cos x


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


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


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


Choose the correct alternative from the following.

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


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


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


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


Evaluate `int (2x + 1)/((x + 1)(x - 2))  "d"x`


`int "e"^x x/(x + 1)^2  "d"x`


Evaluate the following:

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


Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`


Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`


Evaluate the following:

`int_0^pi x log sin x "d"x`


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


Find: `int e^x.sin2xdx`


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


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 ______.


Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.


`int1/(x+sqrt(x))  dx` = ______


The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.


Evaluate the following.

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


Evaluate:

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


Evaluate:

`int (logx)^2 dx`


Evaluate the following.

`intx^3  e^(x^2) 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/(sqrt(1 + x^4))dx`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×