e ^ 3 x - e ^ 2 x e ^x+1 d x - Vidyadip

Question

$\int \sqrt{\frac{ e ^{3 x}- e ^{2 x}}{ e ^x+1}} d x$

✓

Answer

$ \text { Let } I =\int \sqrt{\frac{ e ^{3 x}- e ^{2 x}}{ e ^x+1}} d x$
$=\int \sqrt{\frac{ e ^{2 x}\left( e ^x-1\right)}{ e ^x+1}} d x$
$=\int e ^x \sqrt{\frac{ e ^x-1}{ e ^x+1}} d x $
Put $e^x=t$
$ \therefore e^{ x } d x = dt$
$\therefore I =\int \sqrt{\frac{ t -1}{ t +1}} dt$
$=\int \sqrt{\frac{ t -1}{ t +1} \times \frac{ t -1}{ t -1} dt }$
$=\int \frac{ t -1}{\sqrt{ t ^2-1}} dt$
$=\int\left(\frac{ t }{\sqrt{ t ^2-1}}-\frac{1}{\sqrt{ t ^2-1}}\right) dt$
$=\int \frac{ t }{\sqrt{ t ^2-1}} dt -\int \frac{1}{\sqrt{ t ^2-1}} dt $
$ =I_1-I_2 \quad \ldots \ldots . .(i)$
$I_1=\int \frac{t}{\sqrt{t^2-1}} d t $
Put $t^2-1=a$
$ \therefore 2 t d t = da$
$\therefore I _1=\frac{1}{2} \int \frac{ da }{\sqrt{ a }}$
$=\frac{1}{2} \int a ^{\frac{1}{2}} da$
$=\frac{1}{2}\left(\frac{ a ^{\frac{1}{2}}}{\frac{1}{2}}\right)+ c _1$
$=\sqrt{ a }+ c _1$
$=\sqrt{ t ^2-1}+ c _1$
$\therefore I _1=\sqrt{ e ^{2 x}-1}+ c _1 \quad \ldots \ldots . \text { (ii) }$
$I _2=\int \frac{1}{\sqrt{ t ^2-1^2}} dt$
$=\log \left| t +\sqrt{ t ^2-1^2}\right|+ c _2$
$=\log \left| e ^x+\sqrt{ e ^{2 x}-1}\right|+ c _2\ldots(iii) $
From (i), (ii) and (iii), we get
$\left|=\sqrt{ e ^{2 x}-1}-\log \right| e ^x+\sqrt{ e ^{2 x}-1} \mid+ c _t$
where $c = c _1- c _2$

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 STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Integrate the following w.r.t. x:

$\log (\log x)+(\log x)^{-2}$

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

Evaluate the following:

$\int_0^\pi x \cdot \sin x \cdot \cos ^4 x d x$

Solve the following system of equations by using inversion method $ x+y=1, y+z=\frac{5}{3}, z+x=\frac{4}{3} $

Show that $\frac{d y}{d x}=\frac{y}{x}$ in the following, where a and $p$ are constants.

$\sin \left(\frac{x^3-y^3}{x^3+y^3}\right)=a^3$

If $y=f(u)$ is a differentiable function of $u$ and $u= g (x)$ is a differentiable function of $x$ such that the composite function $y=f(g(x))$ is a differentiable function of $x$ then prove that: $\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}$
Hence find $\frac{d}{d x}\left(\frac{1}{\sqrt{\sin x}}\right)$.

Find the volume of a tetrahedron whose vertices are $A (- 1, 2, 3), B (3, - 2, 1), C (2, 1, 3)$ and $D (- 1, 2, 4).$

D and E divides sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE

The following is the $\text{c.d.f.}$ of $\text{r.v. X:}$

$X$	$−3$	$−2$	$−1$	$0$	$1$	$2$	$3$	$4$
$F(X)$	$0.1$	$0.3$	$0.5$	$0.65$	$0.75$	$0.85$	$0.9$	$1$

Find $\text{p.m.f.}$ of $X.$
$i. P(–1 \leq X \leq 2)$
$ii. P(X \leq 3 / X > 0).$

Evaluate the following integrals as a limit of a sum.

$\int_0^2\left(3 x^2-1\right) d x$