The value of int_{-frac{pi}{2}}^{frac{pi}{2}} | vedclass

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

Question

(C) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$. Since $-\frac{\pi}{2} \approx -1.57$ and $\frac{\pi}{2} \approx 1.57$, we spli

Vedclass · Accepted Answer

$\frac{7}{60}(3\pi-1)$

Vedclass · Answer

(C) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$. Since $-\frac{\pi}{2} \approx -1.57$ and $\frac{\pi}{2} \approx 1.57$, we split the integral based on the values of $[x]$: $I = \int_{-\frac{\pi}{2}}^{-1} \frac{1}{[x]+4} dx + \int_{-1}^{0} \frac{1}{[x]+4} dx + \int_{0}^{1} \frac{1}{[x]+4} dx + \int_{1}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$ For $x \in [-\frac{\pi}{2}, -1)$, $[x] = -2$. For $x \in [-1, 0)$, $[x] = -1$. For $x \in [0, 1)$, $[x] = 0$. For $x \in [1, \frac{\pi}{2}]$, $[x] = 1$. $I = \int_{-\frac{\pi}{2}}^{-1} \frac{1}{-2+4} dx + \int_{-1}^{0} \frac{1}{-1+4} dx + \int_{0}^{1} \frac{1}{0+4} dx + \int_{1}^{\frac{\pi}{2}} \frac{1}{1+4} dx$ $I = \frac{1}{2} [-1 - (-\frac{\pi}{2})] + \frac{1}{3} [0 - (-1)] + \frac{1}{4} [1 - 0] + \frac{1}{5} [\frac{\pi}{2} - 1]$ $I = \frac{1}{2} (\frac{\pi}{2} - 1) + \frac{1}{3} (1) + \frac{1}{4} (1) + \frac{1}{5} (\frac{\pi}{2} - 1)$ $I = \frac{\pi}{4} - \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{\pi}{10} - \frac{1}{5}$ $I = (\frac{\pi}{4} + \frac{\pi}{10}) + (-\frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5})$ $I = \frac{7\pi}{20} + \frac{-30+20+15-12}{60} = \frac{7\pi}{20} - \frac{7}{60} = \frac{21\pi - 7}{60} = \frac{7}{60}(3\pi - 1)$

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$, where $[\bullet]$ denotes the greatest integer function, is

Similar Questions

Given $\int_{0}^{\frac{\pi}{2}} \frac{dx}{1 + \sin x + \cos x} = \ln 2$,then the value of the definite integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \sin x + \cos x} dx$ is equal to

The value of $\int_{\pi /4}^{3\pi /4} \frac{\phi}{1 + \sin \phi} \, d\phi$ is

If $f(a + b - x) = f(x),$ then $\int_{a}^{b} x f(x) dx$ is equal to

If $a=2n$ and $b=2m+1$ for all $m, n \in N$,then evaluate the integral: $\int_{-\pi}^{\pi} e^{\sin^a x} \cot^b((2n+1)x) dx$.

If $f$ and $g$ are continuous functions in $[0, a]$ satisfying $f(x) = f(a - x)$ and $g(x) + g(a - x) = 4$,then $\int_{0}^{a} f(x) g(x) dx$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module