समाकलन int_{frac{pi }{12}}^{frac{pi }{4}} fra | vedclass

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

Question

(A) हम जानते हैं कि $	an x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \co

Vedclass · Accepted Answer

$\frac{15}{128}$

Vedclass · Answer

(A) हम जानते हैं कि $	an x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x}$.इस मान को समाकलन में रखने पर:$I = \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} \frac{8 \cos 2x}{(\frac{2}{\sin 2x})^3} dx = \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} \frac{8 \cos 2x \sin^3 2x}{8} dx = \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} \cos 2x \sin^3 2x dx$.माना $u = \sin 2x$,तब $du = 2 \cos 2x dx$,अर्थात $\cos 2x dx = \frac{du}{2}$.जब $x = \frac{\pi}{12}$,तब $u = \sin(\frac{\pi}{6}) = \frac{1}{2}$.जब $x = \frac{\pi}{4}$,तब $u = \sin(\frac{\pi}{2}) = 1$.$I = \int_{1/2}^{1} u^3 \frac{du}{2} = \frac{1}{2} [\frac{u^4}{4}]_{1/2}^{1} = \frac{1}{8} [1^4 - (\frac{1}{2})^4] = \frac{1}{8} [1 - \frac{1}{16}] = \frac{1}{8} 	imes \frac{15}{16} = \frac{15}{128}$.

समाकलन $\int_{\frac{\pi }{12}}^{\frac{\pi }{4}} \frac{8 \cos 2x}{(\tan x + \cot x)^3} dx$ का मान ज्ञात कीजिए।

Similar Questions

$\int\limits_1^{\sqrt 2 } {\frac{{{x^2} + 1}}{{{x^4} + 1}}} \,dx$ का मान ज्ञात कीजिए:

निश्चित समाकलन का मान ज्ञात कीजिए: $\int_0^{\pi /2} \cos^2 x \, dx$

समाकल $\int_{1}^{5}[|x-3|+|1-x|] dx$ का मान किसके बराबर है?

माना $\alpha > 0$ है। यदि $\int \limits _0^\alpha \frac{ x }{\sqrt{ x +\alpha}-\sqrt{ x }} dx =\frac{16+20 \sqrt{2}}{15}$ है,तो $\alpha$ का मान ज्ञात कीजिए:

Vedclass Test Series

Exam Paper Generator

Online Exam Module