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

MCQ

$\int_{}^{} {\tan x} {\sec ^2}x\sqrt {1 - {{\tan }^2}x} \;dx = $

✓
$ - \frac{1}{3}{(1 - {\tan ^2}x)^{3/2}} + c$
B
$\frac{1}{3}{(1 - {\tan ^2}x)^{3/2}} + c$
C
$ - \frac{2}{3}{(1 - {\tan ^2}x)^{2/3}} + c$
D
None of these

✓

Answer

Correct option: A.

$ - \frac{1}{3}{(1 - {\tan ^2}x)^{3/2}} + c$

a
(a)$\int_{}^{} {\tan x\,.\,{{\sec }^2}x\sqrt {1 - {{\tan }^2}x} \,dx} $
Put $\tan x = t \Rightarrow {\sec ^2}x\,dx = dt,$ then it reduces to $\int_{}^{} {t\sqrt {1 - {t^2}} \,dt} $
Now again, put $1 - {t^2} = u,$ then its reduced form is $ - 2tdt = du$
$ - \frac{1}{2}\int_{}^{} {\sqrt u \,du} = - \frac{1}{3}{u^{3/2}} + c = - \frac{1}{3}{(1 - {\tan ^2}x)^{3/2}} + 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 the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

If $G$ be the geometric mean of $x$ and $y$, then $\frac{1}{{{G^2} - {x^2}}} + \frac{1}{{{G^2} - {y^2}}} = $

If the area of the triangle formed by the positive $x-$axis, the normal and the tangent to the circle $(x-2)^{2}+(y-3)^{2}=25$ at the point $(5,7)$ is $A$ then $24 A$ is equal to ...... .

On the ellipse $4{x^2} + 9{y^2} = 1$, the points at which the tangents are parallel to the line $8x = 9y$ are

If $\mathrm{a, b, c}$ are in $\mathrm{A.P}$, find value of

$\left|\begin{array}{ccc}
2 y+4 & 5 y+7 & 8 y+a \\
3 y+5 & 6 y+8 & 9 y+b \\
4 y+6 & 7 y+9 & 10 y+c
\end{array}\right|$

The greatest integer less than or equal to $\int_1^2 \log _2\left(x^3+1\right) d x+\int_1^{\log 2}\left(2^x-1\right)^{\frac{1}{3}} d x$ is. . . . .

Let $n$ be a positive integer such that

$\log _2 \log _2 \log _2 \log _2 \log _2(n)<0<\log _2 \log _2 \log _2 \log _2(n)$. Let $l$ be the number of digits in the binary expansion of $n$. Then the minimum and the maximum possible values of $l$ are

The value of ${\cos ^2}\,{10^o}\,\, - \,\cos \,\,{10^o}\,\cos \,\,{50^o}\, + \,{\cos ^2}\,{50^o}$ is

If vertices of a parallelogram are respectively $(0, 0)$, $(1, 0)$, $(2, 2)$ and $(1, 2)$, then angle between diagonals is

The solution of the equation ${\log _7}{\log _5}$ $(\sqrt {{x^2} + 5 + x} ) = 0$