int_{0}^{frac{pi}{2}} frac{sin^{frac{2}{3}} x | vedclass

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

Question

(A) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ गुणधर्म $\int_{0}^{a}

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ गुणधर्म $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ का उपयोग करने पर,हमें प्राप्त होता है: $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x)}{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x) + \cos^{\frac{2}{3}}(\frac{\pi}{2}-x)} dx$ चूंकि $\sin(\frac{\pi}{2}-x) = \cos x$ और $\cos(\frac{\pi}{2}-x) = \sin x$,इसलिए: $I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^{\frac{2}{3}} x}{\cos^{\frac{2}{3}} x + \sin^{\frac{2}{3}} x} dx \quad ...(2)$ समीकरण $(1)$ और $(2)$ को जोड़ने पर: $2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx$ $2I = \int_{0}^{\frac{\pi}{2}} 1 dx$ $2I = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}$ $I = \frac{\pi}{4}$

$\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx =$

Similar Questions

$\int_{- 1 / 2}^{1 / 2} \{ [x] + \log (\frac{1 + x}{1 - x}) \} dx =$

$\int_0^1 \frac{8 \log (1+x)}{1+x^2} dx =$

यदि $M = \int_{0}^{\pi / 2} \frac{\cos x}{x+2} dx$ और $N = \int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{(x+1)^{2}} dx$ है, तो $M-N$ का मान ज्ञात कीजिए।

समाकलन $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x}{1+\sin x} dx$ का मान है

Vedclass Test Series

Exam Paper Generator

Online Exam Module