If int_0^1 f(x) dx = 1,int_0^1 x f(x) dx = a, | vedclass

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

Question

(D) We are given the integrals $\int_0^1 f(x) dx = 1$,$\int_0^1 x f(x) dx = a$,and $\int_0^1 x^2 f(x) dx = a^2$. Expanding the integrand $(x-a)^2 f(x)

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) We are given the integrals $\int_0^1 f(x) dx = 1$,$\int_0^1 x f(x) dx = a$,and $\int_0^1 x^2 f(x) dx = a^2$. Expanding the integrand $(x-a)^2 f(x) = (x^2 - 2ax + a^2) f(x) = x^2 f(x) - 2ax f(x) + a^2 f(x)$. Now,integrate term by term: $\int_0^1 (x-a)^2 f(x) dx = \int_0^1 x^2 f(x) dx - 2a \int_0^1 x f(x) dx + a^2 \int_0^1 f(x) dx$. Substituting the given values: $= a^2 - 2a(a) + a^2(1) = a^2 - 2a^2 + a^2 = 0$.

If $\int_0^1 f(x) dx = 1$,$\int_0^1 x f(x) dx = a$,and $\int_0^1 x^2 f(x) dx = a^2$,then $\int_0^1 (x-a)^2 f(x) dx$ is equal to

Similar Questions

$\int_0^1 x^{5/2} (1-x)^{3/2} \, dx =$

$\int_0^{\pi /2} {\frac{{dx}}{{2 + \cos x}}} = $

$\int_0^\pi \frac{1}{4+3 \cos x} d x=$

If $\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$,then $a$ and $b$ are respectively

$\int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module