_ ^ a - x x ;dx = - Vidyadip

MCQ

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

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

✓

Answer

Correct option: A.

$a\left[ {{{\sin }^{ - 1}}\sqrt {\frac{x}{a}} + \sqrt {\frac{x}{a}} \sqrt {\frac{{a - x}}{a}} } \right] + c$

a
(a) $I = \int_{}^{} {\sqrt {\frac{{a - x}}{x}} \,dx} $.
Put $x = a{\sin ^2}\theta \Rightarrow dx = 2a\sin \theta \cos \theta \,d\theta ,$ then
$I = \int_{}^{} {\sqrt {\frac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }}} } \,.\,2a\sin \theta \cos \theta \,d\theta $
$ = a\int_{}^{} {2{{\cos }^2}\theta \,d\theta } = a\int_{}^{} {(1 + \cos 2\theta )\,d\theta } $
$ = a\,\left[ {{{\sin }^{ - 1}}\sqrt {\frac{x}{a}} + \sqrt {\frac{x}{a}} \,.\,\sqrt {\frac{{a - x}}{a}} } \right] + 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 $c$ is a point at which Rolle's theorem holds for the function, $f(\mathrm{x})=\log _{\mathrm{e}}\left(\frac{\mathrm{x}^{2}+\alpha}{7 \mathrm{x}}\right)$ in the interval $[3,4],$ where $\alpha \in \mathrm{R},$ then $f^{\prime \prime}(\mathrm{c})$ is equal to

The normal at the point (1, 1) on the curve 2y + x² = 3 is:

x + y = 0
x - y = 0
x + y + 1 = 0
x - y = 1

If A is a matrix of order m×n and B is a matrix such that AB^T and B^TA are both defined, then the order of matrix B is:

m×n
n×n
n×m
m×n

Let $A = \left\{ {\left( {x,y} \right):{y^2} \le 4x,y - 2x \ge - 4} \right\}$ .The area of the region $A$ is

If $f ( a + b - x )= f ( x ),$ then $\int_{a}^{b} x f(x) d x$ is equal to

A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the sane colour is.

Let $f(x)=2+\cos x$ for all real $x$.

$STATEMENT -1$ : For each real $\mathrm{t}$, there exists a point $\mathrm{c}$ in $[\mathrm{t}, \mathrm{t}+\pi]$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$. because

$STATEMENT -2$: $f(t)=f(t+2 \pi)$ for each real $t$.

$\int_{}^{} {{e^{ - x}}{\rm{cose}}{{\rm{c}}^2}(2{e^{ - x}} + 5)} \;dx = $

The length of the $\perp$

0
$2\sqrt{3}$
$\frac{2}{3}$
2

If $2 x^y+3 y^x=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to