1+x x+ e ^ -x d x - Vidyadip

Question

$\int \frac{1+x}{x+ e ^{-x}} d x$

✓

Answer

$ \text { Let } I =\int \frac{1+x}{x+ e ^{-x}} d x$
$=\int \frac{1+x}{x+\frac{1}{ e ^x}} d x$
$=\int \frac{1+x}{\frac{x \cdot e ^x+1}{ e ^x}} d x$
$=\int \frac{ e ^x(1+x)}{x \cdot e ^x+1} d x $
Put $x \cdot e^x+1=t$
$ \therefore\left[x \cdot\left(e^x\right)+e^x \cdot(1)+0\right] d x=d t$
$\therefore e^x(x+1) d x=d t$
$\therefore I=\int \frac{d t}{t}$
$=\log |t|+c $
$\therefore I =\log \left| x \cdot e ^{ x }+1\right|+ c$

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

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.

Fit a trend line to data in Problem 10 by the method of least squares.

Evalute : $\int \frac{5 x^2-6 x+3}{2 x-3} d x$

For each of the following differential equations find the particular solution : $\left(x-y^2 x\right) d x-\left(y+x^2 y\right) d y=0$, when $x=2, y=0$.

The equations of the two lines of regression are $6x + y − 31 = 0$ and $3x + 2y – 26 = 0.$ Find the value of the correlation coefficient

x	y	xy	$x ^2$	$y^2$
6	9	54	36	81
2	11	22	4	121
10	5	50	100	25
4	8	32	16	64
8	7	$\square$	64	49
Total = 30	Total = 40	Total =$\square$	Total = 220	Total =$\square$

$ b_{x y}=\frac{\square}{\square}$
$b_{y x}=\frac{\square}{\square} $
$\therefore$ Regression equation of $x$ on $y$ is$\square$
$\therefore$ Regression equation of $y$ on $x$ is $\square$

Evaluate $\int_0^{ a } x^2( a -x)^{\frac{3}{2}} d x$

Calculate
a) Laspeyre’s
b) Passche’s
c) Dorbish-Bowley’s Price Index Numbers for following data.

Commodity	Base Year		Current Year
Commodity	Price	Quantity	Price	Quantity
A	10	9	50	8
B	20	5	60	4
C	30	7	70	3
D	40	8	80	2

Find the area of the circle $x^2 + y^2 = 16$

Let the p.m.f. of the r.v. X be

$p(x)= \begin{cases}\frac{3-x}{10} & \text { for } x=-1,0,1,2 \\ 0 & \text { otherwise }\end{cases}$

Calculate E(X) and Var(X).