_ ^ a^x 1 - a^ 2x dx =

MCQ

$\int_{}^{} {\frac{{{a^x}}}{{\sqrt {1 - {a^{2x}}} }}dx = } $

✓
$\frac{1}{{\log a}}{\sin ^{ - 1}}{a^x} + c$
B
${\sin ^{ - 1}}{a^x} + c$
C
$\frac{1}{{\log a}}{\cos ^{ - 1}}{a^x} + c$
D
${\cos ^{ - 1}}{a^x} + c$

✓

Answer

Correct option: A.

$\frac{1}{{\log a}}{\sin ^{ - 1}}{a^x} + c$

a
(a) Put ${a^x} = t \Rightarrow {a^x}{\log _e}a\,dx = dt,$ then
$\int_{}^{} {\frac{{{a^x}}}{{\sqrt {1 - {a^{2x}}} }}\,dx = \frac{1}{{{{\log }_e}a}}\int_{}^{} {\frac{{dt}}{{\sqrt {1 - {t^2}} }}} } $
$ = \frac{1}{{{{\log }_e}a}}{\sin ^{ - 1}}(t) + c = \frac{{{{\sin }^{ - 1}}({a^x})}}{{{{\log }_e}a}} + 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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $a_1, a_2, a_3 …………$ an are in $A.P$ and $a_1 + a_4 + a_7 + …………… + a_{16} = 114$, then $a_1 + a_6 + a_{11} + a_{16}$ is equal to

→

The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median is of length $3a$ is

→

Let $P=\left[\begin{array}{ccc}-30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2 & 7 & \omega^{2} \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1\end{array}\right]$

where $\omega=\frac{-1+ i \sqrt{3}}{2},$ and $I _{3}$ be the identity matrix of order $3$. If the determinant of the matrix $\left( P ^{-1} AP - I _{3}\right)^{2}$ is $\alpha \omega^{2},$ then the value of $\alpha$ is equal to

→

Number of functions $f:[0,1] \rightarrow[0,1]$ satisfying $|f(x)-f(y)|=|x-y|$ for all $x, y$ in $[0,1]$ is

→

The differential equation $\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}$ determines a family of circles with

→

Let $x > 0$ be a fixed real number. Then, the integral $\int \limits_0^{\infty} e^{-t}|x-t| d t$ is equal to

→

How many triangles can be formed by joining four points on a circle

→

The maximum value of ${\left( {\frac{1}{x}} \right)^{2{x^2}}}$ is

→

Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $R$ is: