x (1+ x)^2 d x - Vidyadip

Question

$\int \frac{\log x}{(1+\log x)^2} d x$

✓

Answer

Let $I =\int \frac{\log x}{(1+\log x)^2} d x$
Put $\log x=t$
$ \therefore x = e ^{ t }$
$\therefore d x= e ^{ t } dt$
$\therefore I =\int \frac{ t }{(1+ t )^2} e ^{ t } dt$
$=\int e ^{ t }\left[\frac{( t +1)-1}{(1+ t )^2}\right] dt$
$=\int e ^{ t }\left[\frac{ t +1}{(1+ t )^2}-\frac{1}{(1+ t )^2}\right] dt$
$=\int e ^{ t }\left[\frac{1}{1+ t }-\frac{1}{(1+ t )^2}\right] dt $
$\text { Put } f ( t )=\frac{1}{1+ t }$
$\therefore f ^{\prime}( t )=\frac{-1}{(1+ t )^2}$
$\therefore I =\int e ^{ t }\left[ f ( t )+ f ^{\prime}( t )\right] dt$
$= e ^{ t } f ( t )+ c$
$= e ^{ t } \cdot \frac{1}{1+ t }+ c$
$\therefore I =\frac{x}{1+\log x}+ c $

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(4 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Find Price Index Number using Simple Aggregate method by taking $2000$ as base year

Commodity	Price (in ₹) for year 2000	Price (in ₹) for year 2007
Watch	900	1,475
Shoes	1,760	2,300
Sunglasses	60	1,040
Mobile	4,500	8,500

The equations of the two lines of regression are 6x + y − 31 = 0 and 3x + 2y – 26 = 0. Calculate the mean values of x and y

Solve the following differential equation
$\sec ^2 x \tan y d x+\sec ^2 y \tan x d y=0$
Solution: $\sec ^2 x \tan y d x+\sec ^2 y \tan x d y=0$
$\therefore \frac{\sec ^2 x}{\tan x} d x+\square=0$
Integrating, we get
$\square+\int \frac{\sec ^2 y}{\tan y} d y=\log c$
Each of these integral is of the type
$\int \frac{ f ^{\prime}(x)}{ f (x)} d x=\log | f ( x )|+\log C$
$\therefore$ the general solution is
$\square+\log |\tan y|=\log c$
$\therefore \log |\tan x \cdot \tan y|=\log c$
$\square$
This is the general solution.

Evaluate $\int_2^3 \frac{x}{(x+2)(x+3)} d x$

Find the population of city at any time $t$ given that rate of increase of population is proportional to the population at that instant and that in a period of 40 years the population increased from 30000 to 40000 .
Solution: Let $p$ be the population at time $t$.
Then the rate of increase of $p$ is $\frac{d p}{d t}$ which is proportional to $p$.
$\therefore \frac{ dp }{ dt } \propto p$
$\therefore \frac{ dp }{ dt }= kp$, where $k$ is a constant.
$\therefore \frac{ dp }{ p }= kdt$
On integrating, we get
$ \int \frac{ dp }{ p }= k \int dt$
$\therefore \log p = kt + c $
Initially, i.e. when $t=0$, let $p=30000$
$\therefore \log 30000= k \times 0+ c$
$\therefore c =\square$
$\therefore \log p=k t+\log 30000$
$\therefore \log p-\log 30000=k t$
$\therefore \log \left(\frac{ p }{30000}\right)= kt\ldots(i)$
when $t=40, p=40000$
$ \therefore \log \left(\frac{40000}{30000}\right)=40 k$
$\therefore k =\square $
$\therefore$ equation (1) becomes, $\log \left(\frac{p}{30000}\right)=\square$
$\therefore \log \left(\frac{ p }{30000}\right)=\frac{ t }{40} \log \left(\frac{4}{3}\right)$
$\therefore p=\square$

Calculate Quantity Index Number using Simple Aggregate method

Commodity	I	II	III	IV	V
Base year Quantity	140	120	100	200	225
Current year Quantity	100	80	70	150	185

Minimize Z = 2x + 3y subject to constraintsx + y ≥ 6, 2x + y ≥ 7, x + 4y ≥ 8, x ≥ 0, y ≥ 0

Four new machines $M_1, M_2, M_3$, and $M_4$ are to be installed in a machine shop. There are five vacant places $A, B, C, D,$ and $E$ available. Because of limited space, machine $M_2$ cannot be placed at $C$ and $M_1$ cannot be placed at $A$. The cost matrix is given below.

Machines	Places
	A	B	C	D	E
M1	$4$	$6$	$10$	$5$	$6$
M2	$7$	$4$	-	$5$	$4$
M3	-	$6$	$9$	$6$	$2$
M4	$9$	$3$	$7$	$2$	$3$

Find the optimal assignment schedule.

Write the dual statement of each of the following compound statements:
(i) 13 is prime number and India is a democratic country.
(ii) Karina is very good or everybody likes her.
(iii) Radha and Sushmita can not read Urdu.
(iv) A number is real number and the square of the number is non-negative.

If $x + y = 3$ show that the maximum value of $x^2y$ is $4.$