_ ^ x 1 - x^2 1 + x^2 ;dx = - Vidyadip

MCQ

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

✓
$\frac{1}{2}[{\sin ^{ - 1}}{x^2} + \sqrt {1 - {x^4}} ] + c$
B
$\frac{1}{2}[{\sin ^{ - 1}}{x^2} + \sqrt {1 - {x^2}} ] + c$
C
${\sin ^{ - 1}}{x^2} + \sqrt {1 - {x^4}} + c$
D
${\sin ^{ - 1}}{x^2} + \sqrt {1 - {x^2}} + c$

✓

Answer

Correct option: A.

$\frac{1}{2}[{\sin ^{ - 1}}{x^2} + \sqrt {1 - {x^4}} ] + c$

a
(a)$\int_{}^{} {x\sqrt {\frac{{1 - {x^2}}}{{1 + {x^2}}}} } dx = \int_{}^{} {\frac{{x.\,(1 - {x^2})}}{{\sqrt {1 - {x^4}} }}} dx$
$\{$ Multiplying ${N^r}$ and ${D^r}$ by ${(1 - {x^2})^{1/2}}\} $
$ = \int_{}^{} {\frac{x}{{\sqrt {1 - {x^4}} }}} \,dx - \int_{}^{} {\frac{{{x^3}}}{{\sqrt {1 - {x^4}} }}} dx$

$ = \frac{1}{2}[{\sin ^{ - 1}}({x^2}) + \sqrt {1 - {x^4}} ] + c$.
(By putting ${x^2} = t$ and $\sqrt {1 - {x^4}} = \sqrt t $respectively)

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

The value of $\sum\limits_{r = 16}^{30} {(r + 2)(r - 3)}$ is equal to

In a $\triangle A B C$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x-y=2$. If $2 A B=B C$ and the point $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$, then $\alpha+2 \beta$ is equal to

The greatest and the least value of ${({\sin ^{ - 1}}x)^3} + {({\cos ^{ - 1}}x)^3}$ are

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\, - {x^2},\,{\rm{when\,\, }}x \le 0\\\,\,\,\,\,5x - 4,\,{\rm{when\,\,}}0 < x \le 1\\4{x^2} - 3x,\,{\rm{when\,\, }}1 < x < 2\\\,\,\,\,\,3x + 4,{\rm{when \,\,}}x \ge 2\end{array} \right.$, then

$f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x$

Which term of the sequence $( - 8 + 18i),\,( - 6 + 15i),$ $( - 4 + 12i)$ $,......$ is purely imaginary

$P$ is a point on the parabola $y^2 = 4ax$ $ (a > 0)$ whose vertex is $ A$ . $PA $ is produced to meet the directrix in $D$ and $M$ is the foot of the perpendicular from $ P$ on the directrix. If a circle is described on $MD$ as a diameter then it intersects the $x-$ axis at a point whose co-ordinates are :

Let $S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}$, where $i=\sqrt{-1} .$ Then the number of $2 -$ digit numbers in the set $\mathrm{S}$ is $......$

$\int {\cos ({{\log }_e}x)\,dx} $ is equal to

Domain of $log\,log\,log\, ....(x)$ is

$ \leftarrow \,n\,\,times\, \to $