e^x - 1 dx = - Vidyadip

MCQ

$\int {\sqrt {{e^x} - 1} } dx = $

✓
$2\left[ {\sqrt {{e^x} - 1} - {{\tan }^{ - 1}}\sqrt {{e^x} - 1} } \right] + c$
B
$\sqrt {{e^x} - 1} - {\tan ^{ - 1}}\sqrt {{e^x} - 1} + c$
C
$\sqrt {{e^x} - 1} + {\tan ^{ - 1}}\sqrt {{e^x} - 1} + c$
D
$2\left[ {\sqrt {{e^x} - 1} + {{\tan }^{ - 1}}\sqrt {{e^x} - 1} } \right] + c$

✓

Answer

Correct option: A.

$2\left[ {\sqrt {{e^x} - 1} - {{\tan }^{ - 1}}\sqrt {{e^x} - 1} } \right] + c$

a
(a)$A = \int {\sqrt {{e^x} - 1} } \,dx$
Let ${e^x} - 1 = {t^2}$ ==> ${e^x}dx = 2t\,dt$. Hence

$dx = \frac{{2t}}{{{t^2} + 1}}dt$
$A = \int {t\frac{{2t}}{{{t^2} + 1}}} dt = \int {\frac{{2{t^2}}}{{{t^2} + 1}}dt} $
$ = \int {\frac{{2({t^2} + 1) - 2}}{{{t^2} + 1}}dt} $$ = \int {2dt - \int {\frac{{2dt}}{{{t^2} + 1}}} } $
$ = 2t - 2{\tan ^{ - 1}}t + c = 2\sqrt {{e^x} - 1} - 2{\tan ^{ - 1}}\sqrt {{e^x} - 1} + 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 "M.C.Q (1 Marks)" from 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

0.32
0.032
5.68
5.968

If $f(x) = \left\{ \begin{array}{l}\frac{{{x^2} - 1}}{{x + 1}},\,{\rm{when \,\,}}x \ne - 1\\\,\,\,\,\,\,\,\, - 2,\,{\rm{when\,\, }}x = - 1\end{array} \right.$, then

The equation of the curve whose slope is given by $\frac{\text{dy}}{\text{dx}}=\frac{2\text{y}}{\text{x}};\text{x}>0,\text{y}>0$ and which passes through the point (1, 1) is:

$\text{x}^{2}=\text{y}$
$\text{y}^{2}=\text{x}$
$\text{x}^{2}=2\text{y}$
$\text{y}^{2}=2\text{x}$

If the matrix A is both symmetric and skew symmetric, then:

A is a diagonal matrix.
A is a zero matrix.
A is a square matrix.
None of these.

If $\Delta=\left|\begin{array}{ccc}l+m & m+n & n+l \\ n & l & m \\ 2 & 2 & 2\end{array}\right|$, then

$\left| {\,\begin{array}{*{20}{c}}{19}&{17}&{15}\\9&8&7\\1&1&1\end{array}\,} \right| = $

Let $f( x)$ be a polynomial of degree four having extreme values at $ x=1 $ and $ x=2$ . If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f\left( x \right)}}{{{x^2}}}} \right] = 3$,then $f\left( 2 \right)$ is equal to :

Define a function $f: R \rightarrow R$ by $f(x)=\left\{\begin{array}{cc}\frac{\sin x^2}{x}, & \text { for } x<0 \\ x^2+a x+b, & \text { for } x \geq 0\end{array}\right.$ Suppose $f(x)$ is differentiable of $R$. Then,

$\tan \left[ {2{{\tan }^{ - 1}}\left( {\frac{1}{5}} \right) - \frac{\pi }{4}} \right] = $

Let $f(x) = \left\{ \begin{array}{l}{(1 + |\sin x|)^{a/|\sin x|}},\,\, - \pi /6 < x < 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b,\,x = 0\\{e^{\tan 2x/\tan 3x}},\,0 < x < - \pi /6\end{array} \right.$ then the value of $a$ and $b$ if $f$ is continuous at $x = 0$, are respectively