x 1-x dx is equal to: - Vidyadip

MCQ

$\int\sqrt{\frac{\text{x}}{1-\text{x}}}\text{ dx}$ is equal to:

A
$\sin^{-1}\sqrt{\text{x}}+\text{C}$
B
$\sin^{-1}\Big\{\sqrt{\text{x}}-\sqrt{\text{x}(1-\text{x})}\Big\}+\text{C}$
C
$\sin^{-1}\Big\{\sqrt{\text{x}(1-\text{x})}\Big\}+\text{C}$
✓
$\sin^{-1}\sqrt{\text{x}}-\sqrt{\text{x}(1-\text{x})}+\text{C}$

✓

Answer

Correct option: D.

$\sin^{-1}\sqrt{\text{x}}-\sqrt{\text{x}(1-\text{x})}+\text{C}$

$\text{I}=\int\sqrt{\frac{\text{x}}{1-\text{x}}}\text{ dx}$
$\text{I}=\int\sqrt{\frac{\text{x}}{1-\text{x}}\cdot\frac{\text{x}}{\text{x}}}\text{ dx}$
$\text{I}=\int\frac{\text{x dx}}{\sqrt{\text{x}-\text{x}^2}}$
Consider,
$\text{x}=\text{A}\frac{\text{d}(\text{x}-\text{x}^2)}{\text{dx}}+\text{B}$
$\text{x}=\text{A}(1-2\text{x})+\text{B}$
$\text{x}=-2\text{Ax}+\text{A}+\text{B}$
$-2\text{A}=1$
$\text{A}=\frac{-1}{2}$
$\text{I}=\int\frac{\frac{-1}{2}(1-2\text{x})+\frac{1}{2}}{\sqrt{\text{x}-\text{x}^2}}\text{ dx}$
$\text{I}=\int\Big(\frac{-1}{2}\frac{1-2\text{x}}{\sqrt{\text{x}-\text{x}^2}}+\frac{1}{2\sqrt{\text{x}-\text{x}^2}}\Big)\text{dx}$
$\text{I}=\frac{-1}{2}\times2\sqrt{\text{x}-\text{x}^2}+\frac{1}{2}\int\frac{1}{\sqrt{\text{x}-\text{x}^2}}\text{ dx}$
Second term after completing square method you will get as
$\text{I}=-\sqrt{\text{x}-\text{x}^2}+\sin^{-1}\sqrt{\text{x}}+\text{C}$

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 "MCQ" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate: $\int \frac{x^3+x}{x^4-9} d x$

If the points $P (\overline{ a }+2 \overline{b}+\overline{ c }), Q (2 \overline{ a }+3 \overline{b})$ and $R (\overline{ b }+ t \overline{ c })$ are collinear, where $\overline{ a }, \overline{ b }, \overline{ c }$ are three non-coplanar vectors, then the value of $t$ is

If $\bar{a}, \bar{b}, \bar{c}$ are non-collinear vectors such that for some scalars $x, y, z , x \overline{ a }+y \overline{b}+ z \overline{ c }=0$, then

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is $60^{\circ}$ If the third side is 3 , the remaining fourth side is

If $\text{A}=\begin{bmatrix}\alpha&\beta\\\gamma&-\alpha\end{bmatrix}$ is such that $A^2 = I$, then:

Given $X \sim B (n, p)$. If $p=0.6$ and $E ( X )=6$, then the value of Var. $( X )$ is......

If the points $A , B , C$ and D have position vectors $\bar{a}, 2 \bar{a}+\bar{b}, 4 \bar{a}+2 \bar{b}$ and $5 \bar{a}+4 \bar{b}$ respectively, then three collinear points are

If $\int_0^a \sqrt{\frac{a-x}{x}} d x=\frac{K}{2}$, then $K=$

A bag contains $5$ red and $3$ blue balls. If $3$ balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two unit vectors inclined at an angle $\theta$, such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|<1,$ then: