What is the value of int limits_0^1 cos (pi x | vedclass

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

Question

(C) Let $I = \int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$.Since $[2x]$ is a step function,we split the integral at $x = \frac{1}{2}$:For $0 \le

Vedclass · Accepted Answer

$\frac{2}{\pi}$

Vedclass · Answer

(C) Let $I = \int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$.Since $[2x]$ is a step function,we split the integral at $x = \frac{1}{2}$:For $0 \le x For $\frac{1}{2} \le x Thus,$I = \int \limits_0^{1/2} \cos(\pi x) \cdot (1) d x + \int \limits_{1/2}^1 \cos(\pi x) \cdot (-1) d x$.$I = \left[ \frac{\sin(\pi x)}{\pi} \right]_0^{1/2} - \left[ \frac{\sin(\pi x)}{\pi} \right]_{1/2}^1$.$I = \frac{1}{\pi} [\sin(\frac{\pi}{2}) - \sin(0)] - \frac{1}{\pi} [\sin(\pi) - \sin(\frac{\pi}{2})]$.$I = \frac{1}{\pi} [1 - 0] - \frac{1}{\pi} [0 - 1] = \frac{1}{\pi} + \frac{1}{\pi} = \frac{2}{\pi}$.

What is the value of $\int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$? (Here $[t]$ denotes the greatest integer function of the real number $t$.)

Similar Questions

If $\int_1^2 \frac{dx}{(x^2-2x+4)^{\frac{3}{2}}} = \frac{k}{k+5}$,then $k$ has the value

The value of the integral $\int_0^4 \frac{d x}{1+x^2}$ obtained by using the Trapezoidal rule with $h=1$ is

Evaluate the definite integral $\int_2^5 2[x] \, dx$,where $[x]$ denotes the greatest integer function $\leq x$.

$\int_0^2 |2x - 3| \, dx = $

The points of extrema of $f(x) = \int_0^x \frac{\sin t}{t} dt$ in the domain $x > 0$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module