If _ - ^f(x)dx = 1 then _ - ^ f ( x - 1 x )dx is equal to - Vidyadip

MCQ

If $\int\limits_{ - \infty }^\infty {f(x)dx = 1} $ then $\int\limits_{ - \infty }^\infty {f\left( {x - \frac{1}{x}} \right)dx} $ is equal to

A
$0$
✓
$1$
C
$-1$
D
$2$

✓

Answer

Correct option: B.

$1$

b
Put $x=y-\frac{1}{y}$

$\Rightarrow 1 \Rightarrow \int_{-\infty}^{\infty} f\left(y-\frac{1}{y}\right)\left(1+\frac{1}{y^{2}}\right) d y$

$=\int_{-\infty}^{0} f\left(y-\frac{1}{y}\right) d y+\int_{-\infty}^{0} f\left(y-\frac{1}{y}\right) \frac{d y}{y^{2}}$

Putting $z=-\frac{1}{y}=\int_{-\infty}^{0} f\left(y-\frac{1}{y}\right) d y+\int_{0}^{\infty} f\left(z-\frac{1}{z}\right) d z=1$

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

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

Range of $f(x) = sin^{-1} (\sqrt {x^2 + x +1})$ is -

Which of the following functions cannot have their inverse defined ? (where $[.]\, \to$ greatest integer function)

If the function $f\left( x \right) = \left\{ \begin{array}{l} a\,\left| {\pi - x} \right|\, + 1,\,\,x \le 5\,\\ b\,\,\left| {\pi - x} \right|\, + 3,\,\,x > 5\,\, \end{array} \right.$ is continuous at $x = 5$, then the value of $a -b$ is

The arithmetic mean of the nine numbers in the given set $\{9,99,999,...., 999999999\}$ is a $9$ digit number $N$, all whose digits are distinct. The number $N$ does not contain the digit

The points of intersection of the circles ${x^2} + {y^2} = 25$and ${x^2} + {y^2} - 8x + 7 = 0$ are

Let $f: R \rightarrow R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t) d t, x>0$. Then which of the following options is/are correct?

$(1)$ F has a local minimum at $x=1$

$(2)$ $F$ has a local maximum at $x=2$

$(3)$ $F ( x ) \neq 0$ for all $x \in(0,5)$

$(4)$ F has two local maxima and one local minimum in $(0, \infty)$

The value of the integral $\int \limits_{-\pi / 2}^{\pi / 2} \frac{\sin ^2 x}{1+e^x}\,d x$ is

A point equidistant from the lines $4x + 3y + 10 = 0$, $5x - 12y + 26 = 0$ and $7x + 24y - 50 = 0$ is

Let $n \geq 3$. A list of numbers $0 < x_1 < x_2 < \ldots < x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list of numbers is made as follows: $y_1=0, y_2=x_2, \ldots, x_{n-1}$ $=x_n-1, y_n=x_1+x_n$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?