If [cdot] denotes the greatest integer functi | vedclass

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

Question

(D) Let $I = \int_{0}^{\pi} [\cos x] \, dx \quad \dots(1)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0

Vedclass · Accepted Answer

$-\frac{\pi}{2}$

Vedclass · Answer

(D) Let $I = \int_{0}^{\pi} [\cos x] \, dx \quad \dots(1)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0}^{\pi} [\cos(\pi - x)] \, dx = \int_{0}^{\pi} [-\cos x] \, dx \quad \dots(2)$Adding $(1)$ and $(2)$:$2I = \int_{0}^{\pi} ([\cos x] + [-\cos x]) \, dx$Since $[x] + [-x] = -1$ if $x 
otin \mathbb{Z}$ and $0$ if $x \in \mathbb{Z}$,and the set of points where $\cos x \in \mathbb{Z}$ in $[0, \pi]$ is finite (measure zero),we have:$2I = \int_{0}^{\pi} (-1) \, dx$$2I = -[x]_{0}^{\pi} = -\pi$$I = -\frac{\pi}{2}$

If $[\cdot]$ denotes the greatest integer function,then the integral $\int_{0}^{\pi} [\cos x] \, dx$ is equal to:

Similar Questions

$\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x=$

$\int_0^\pi x f(\sin x) dx = $

The absolute value of $\int_{10}^{19} \frac{\sin x}{1 + x^8} dx$ is less than:

Evaluate the integral: $\int_0^{50 \pi} \sqrt{1-\cos 2x} \, dx$ (in $\sqrt{2}$)

For $x > 0,$ if $f(x) = \int_{1}^{x} \frac{\log_{e} t}{1+t} dt,$ then $f(e) + f\left(\frac{1}{e}\right)$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module