If int_{-infty}^{infty} f(x) dx = 1,then int_ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $I = \int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx$. We use the property $\int_{-\infty}^{\infty} g\left(x - \frac{1}{x}\right) dx

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Let $I = \int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx$. We use the property $\int_{-\infty}^{\infty} g\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} g(u) du$ for a suitable transformation. Let $u = x - \frac{1}{x}$. Then $du = (1 + \frac{1}{x^2}) dx$. This integral is a standard result in definite integration involving the substitution $x - \frac{1}{x} = u$. Specifically,$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(u) du$. Given $\int_{-\infty}^{\infty} f(x) dx = 1$,it follows that $I = 1$.

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

Similar Questions

$\int_{0}^{2a} f(x) \, dx = $

$\int_0^{\pi / 2} \log _e(\sin 2 x) d x$

If $h(a) = h(b)$,the value of the integral $\int_a^b {[f(g(h(x)))]^{-1} f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \, dx} = $

$\int\limits_{ - 1}^1 {\frac{{{x^3} + |x| + 1}}{{{x^2} + 2|x| + 1}}} dx = a \ln 2 + b$,then:

$\int_{-1}^1 (a x^3 + b x) dx = 0$ for

Vedclass Test Series

Exam Paper Generator

Online Exam Module