_ ^ x x ^3 x ,dx = - Vidyadip

MCQ

$\int_{}^{} {x\sin x{{\sec }^3}x\,dx = } $

A
$\frac{1}{2}[{\sec ^2}x - \tan x] + c$
✓
$\frac{1}{2}[x{\sec ^2}x - \tan x] + c$
C
$\frac{1}{2}[x{\sec ^2}x + \tan x] + c$
D
$\frac{1}{2}[{\sec ^2}x + \tan x] + c$

✓

Answer

Correct option: B.

$\frac{1}{2}[x{\sec ^2}x - \tan x] + c$

b
(b)$\int_{}^{} {x\sin x{{\sec }^3}x\,dx} = \int_{}^{} {x\sin x\frac{1}{{{{\cos }^3}x}}\,dx} $

$ = \int_{}^{} {x\tan x\,.\,{{\sec }^2}x\,dx} $

Now put $\tan x = t \Rightarrow {\sec ^2}x\,dx = dt$ and $x = {\tan ^{ - 1}}t,$

then it reduces to $\int_{}^{} {{{\tan }^{ - 1}}t\,.\,t\,dt} = \frac{{x{{\tan }^2}x}}{2} - \frac{1}{2}t + \frac{1}{2}{\tan ^{ - 1}}t$

$ = \frac{{x({{\sec }^2}x - 1)}}{2} - \frac{1}{2}\tan x + \frac{1}{2}x = \frac{1}{2}[x{\sec ^2}x - \tan x] + 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

If the line $x = y = z$ intersects the line $x \sin A+ y$ $\sin B + z \sin C -18=0= x \sin 2 A + y \sin 2 B + z$ $\sin 2 C -9$, where $A , B , C$ are the angles of a triangle $ABC$, then $80\left(\sin \frac{ A }{2} \sin \frac{ B }{2} \sin \frac{ C }{2}\right)$ is equal to $..........$.

Choose the correct answer from the given four options.
The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The objective function is Z = 4x + 3y.
Compare the quantity in Column A and Column B.

Column A	Column B
Maximum of Z	325

The quantity in column A is greater .
The quantity in column B is greater.
The two quantities are equal.
The relationship can not be determined on the basis of the information supplied.

The maximum value of the function $f\,(x)\, = 3{x^3} - 18{x^2} + 27x\,\, - 40$ on the set $S = \{ x\, \in \,R\,:\,{x^2}\, + \,30\, \le \,11x\} $ is

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

In a pack of playing cards there are 4 Ace, 4 King, 4 Queen and 4 Jack. Two cards are drawn. What is the probability getting at least one Ace?

The triangle of maximum area that can be inscribed in a given circle of radius $'r'$ is ...... .

If $tan^{-1} (x^2 + 3|x|-4 )= tan^{-1} (4 \pi + sin^{-1}(sin14))$, then the value of $cos^{-1}(cos3|x|)$ is equal to

Let $R=\left(\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right)$ $\neq 0, \theta \in(0,2 \pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of M. Then, among the statements:

$(I)$ $Trace(\mathrm{R})=0$

$(II) $If $trace(\operatorname{adj}(\operatorname{adj}(\mathrm{R}))=0$, then $R$ has exactly one non-zero entry.

The area enclosed by the curve $y = \sqrt x \,\, \&\,\, x = - \sqrt y $ , the circle $x^2 + y^2 = 2$ above the $x-$ axis, is

Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in N$. Then the value of $\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}$ is equal to ...........