Category: Integration

Question Number 54936 by maxmathsup by imad last updated on 14/Feb/19 $$letf\left(\theta \right)={\int }_0^1\sqrt{x^2+2\left(cos\theta \right)x+1}dxwith\theta \in R.$$$$1)calculatef\left(\theta \right)$$$$2)findthevalueofg\left(\theta \right)={\int }_0^1\frac{xsin\theta }{\sqrt{x^2+2cos\theta x+1}}dx$$ Commented…

Question Number 185985 by cortano1 last updated on 30/Jan/23 $$\int \frac{dx}{\sqrt{\sin x(1+\cos x)}}=?$$ Answered by ARUNG_Brandon_MBU last updated on 30/Jan/23 $$I=\int \frac{dx}{\sqrt{\sin x(1+\cos x)}}=\int \frac{dx}{2\sqrt{\sin \frac{x}{2}{\cos }^3\frac{x}{2}}}$$$$\:\:=\int\frac{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\:\sqrt{\mathrm{tan}\frac{{x}}{\mathrm{2}}}}{dx}=\int\frac{{d}\left(\mathrm{tan}\frac{{x}}{\mathrm{2}}\right)}{\:\sqrt{\mathrm{tan}\frac{{x}}{\mathrm{2}}}}=\mathrm{2}\sqrt{\mathrm{tan}\frac{{x}}{\mathrm{2}}}+{C} \…

Question Number 54896 by yusufbode1996 last updated on 14/Feb/19 $$\int \sqrt{}(x2+2x+2)$$ Commented by maxmathsup by imad last updated on 14/Feb/19 $${let}\:{I}\:=\int\:\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{2}}{dx}\:\Rightarrow{I}\:=\int\:\sqrt{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \:+\mathrm{1}}{dx}\:\:{changement}\:{x}+\mathrm{1}\:={sh}\left({t}\right)\:{give} \…

Question Number 185959 by Michaelfaraday last updated on 30/Jan/23 $$solve:$$$$\int x^{Inx}dx$$ Answered by Frix last updated on 30/Jan/23 $$\int{x}^{\mathrm{ln}\:{x}} {dx}\:\overset{{t}=\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{ln}\:{x}} {=}\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{e}}}\int\mathrm{e}^{{t}^{\mathrm{2}}…

Question Number 185935 by cortano1 last updated on 30/Jan/23 $$\int \frac{1-\cos x}{(1+\cos x)\cos x}dx=?$$ Answered by Ar Brandon last updated on 30/Jan/23 $$I=\int \frac{1-\cos x}{(1+\cos x)\cos x}dx=\int \frac{{(1-\cos x)}^2}{(1-{\cos }^{2}x)\cos x}dx$$$$\:\:=\int\frac{\mathrm{1}−\mathrm{2cos}{x}+\mathrm{cos}^{\mathrm{2}}…