Category: Integration

Question Number 219434 by SdC355 last updated on 25/Apr/25 Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 219458 by Nicholas666 last updated on 25/Apr/25 $$$$$$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \int_{\mathrm{1}} ^{\:\mathrm{2}} \:{x}_{\mathrm{1}} ^{\:\mathrm{2}} {x}_{\mathrm{2}} ^{\:\mathrm{3}} {x}_{\mathrm{3}} ^{\:\mathrm{4}}…

Question Number 219384 by mnjuly1970 last updated on 23/Apr/25 Answered by SdC355 last updated on 24/Apr/25 $$\int \frac{\mathrm{d}x}{\sqrt{x}}{\cos }^3\left(x\right)\sin \left(x\right)=I$$$$\int \frac{\mathrm{d}x}{x}\sqrt{x}{\cos }^3\left(x\right)\sin \left(x\right)=\int \frac{\mathrm{d}x}{x}\sqrt{x}{\cos }^2\left(x\right)\cos \left(x\right)\sin \left(x\right)$$$$\int\:\:\frac{\mathrm{d}{x}}{\:\mathrm{2}{x}}\:\sqrt{{x}}\mathrm{cos}^{\mathrm{2}} \left({x}\right)\mathrm{sin}\left(\mathrm{2}{x}\right)=\int\:\:\mathrm{d}{x}\:\frac{{e}^{−{xt}}…

Question Number 219305 by Nicholas666 last updated on 22/Apr/25 $$$$$$Prove;$$$$I_0\left(x\right)=\frac{1}{\pi }{\int }_0^{\pi }{e}^{xcox\left(\theta \right)}d\theta ;$$$$\:\:\:{x}^{\mathrm{2}} {I}_{\mathrm{0}} ^{''} \left({x}\right)\:+\:{xI}'_{\mathrm{0}} \left({x}\right)\:−\:{x}^{\mathrm{2}} {I}_{\mathrm{0}}…

Question Number 219193 by fantastic last updated on 20/Apr/25 $$\int \sqrt{\frac{x+1}{x+2}}.\frac{1}{x+3}dx=?$$ Answered by aleks041103 last updated on 20/Apr/25 $$\frac{x+1}{x+2}=u=1-\frac{1}{x+2}$$$$\Rightarrow x=\frac{1}{1-u}-2=\frac{2u-1}{1-u}$$$$\Rightarrow{dx}=\frac{\mathrm{2}\left(\mathrm{1}−{u}\right)−\left(−\mathrm{1}\right)\left(\mathrm{2}{u}−\mathrm{1}\right)}{\left(\mathrm{1}−{u}\right)^{\mathrm{2}} }{du}=…