यदि int e^xleft(frac{x sin ^{-1} x}{sqrt{1-x^ | vedclass

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

Question

(C) हम जानते हैं कि $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$.माना $f(x) = \frac{x \sin^{-1} x}{\sqrt{1-x^2}}$.तब $f'(x) = \frac{d}{dx} \left( \frac

Vedclass · Accepted Answer

$\frac{\pi}{6} \sqrt{\frac{e}{3}}$

Vedclass · Answer

(C) हम जानते हैं कि $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$.माना $f(x) = \frac{x \sin^{-1} x}{\sqrt{1-x^2}}$.तब $f'(x) = \frac{d}{dx} \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} \right) = \frac{\sqrt{1-x^2} \cdot \frac{d}{dx}(x \sin^{-1} x) - x \sin^{-1} x \cdot \frac{d}{dx}(\sqrt{1-x^2})}{1-x^2}$$= \frac{\sqrt{1-x^2} \left( \sin^{-1} x + \frac{x}{\sqrt{1-x^2}} \right) - x \sin^{-1} x \left( \frac{-x}{\sqrt{1-x^2}} \right)}{1-x^2}$$= \frac{\sqrt{1-x^2} \sin^{-1} x + x + \frac{x^2 \sin^{-1} x}{\sqrt{1-x^2}}}{1-x^2} = \frac{(1-x^2) \sin^{-1} x + x \sqrt{1-x^2} + x^2 \sin^{-1} x}{(1-x^2)^{3/2}} = \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2}$.अतः,समाकलन $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C = e^x \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + C$ है।इसलिए,$g(x) = \frac{x e^x \sin^{-1} x}{\sqrt{1-x^2}}$.$x = \frac{1}{2}$ पर मान रखने पर: $g\left(\frac{1}{2}\right) = \frac{\frac{1}{2} e^{1/2} \sin^{-1}(1/2)}{\sqrt{1-(1/2)^2}} = \frac{\frac{1}{2} \sqrt{e} \cdot \frac{\pi}{6}}{\sqrt{3/4}} = \frac{\frac{\pi \sqrt{e}}{12}}{\frac{\sqrt{3}}{2}} = \frac{\pi \sqrt{e}}{6 \sqrt{3}} = \frac{\pi}{6} \sqrt{\frac{e}{3}}$.

यदि $\int e^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) d x=g(x)+C$ जहाँ $C$ समाकलन स्थिरांक है,तो $g \left(\frac{1}{2}\right)$ का मान ज्ञात कीजिए :

Similar Questions

$\int {\log x(\log x + 2) \, dx} = $

समाकल $\int_{1}^{2} e^{x} \cdot x^{x}(1 + \log_{e} x + 1) dx$ का मान ज्ञात कीजिए।

यदि $I = \int e^{\sin \theta} (\log \sin \theta + \operatorname{cosec}^2 \theta) \cos \theta \, d\theta$ है,तो $I$ का मान ज्ञात कीजिए।

$\int \frac{(x-1) e^x}{(x+1)^3} \,d x$ का मान किसके बराबर है?

$\frac{dy}{dx} = e^x(\sin x + \cos x)$ का हल है

Vedclass Test Series

Exam Paper Generator

Online Exam Module