Category: Integration

Question Number 64068 by mathmax by abdo last updated on 12/Jul/19 $$calculate{\int }_0^{\pi }\frac{tsint}{3+{sin}^{2}t}dt$$ Commented by mathmax by abdo last updated on…

Question Number 129594 by bagjagunawan last updated on 16/Jan/21 Commented by bemath last updated on 17/Jan/21 $$\sin \mathrm{x}+\cos \mathrm{x}+\frac{1+\sin \mathrm{x}+\cos \mathrm{x}}{\cos \mathrm{x}\sin \mathrm{x}}=$$$$\sin \mathrm{x}+\cos \mathrm{x}+\frac{2\cos \left(\frac{\mathrm{x}}{2}\right)(\sin \left(\frac{\mathrm{x}}{2}\right)+\cos \left(\frac{\mathrm{x}}{2}\right))}{2\sin \left(\frac{\mathrm{x}}{2}\right)\cos \left(\frac{\mathrm{x}}{2}\right)\cos \mathrm{x}}=$$$$\sin \mathrm{x}+\cos \mathrm{x}+\frac{1}{\sin \left(\frac{\mathrm{x}}{2}\right)(\cos \left(\frac{\mathrm{x}}{2}\right)-\sin \left(\frac{\mathrm{x}}{2}\right))}=$$$$\sin \mathrm{x}+\cos \mathrm{x}+\frac{2}{\sin \mathrm{x}-1+\cos \mathrm{x}}=$$$$\:\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}+\:\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}−\mathrm{1}}=\frac{\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}}…

Question Number 64037 by MJS last updated on 18/Nov/19 $$\mathrm{reduction}\mathrm{formulas}\mathrm{for}n\in \mathbb{N},\mathrm{some}n>0,\mathrm{some}n>1$$$$$$$$\int {\sin }^{n}xdx=-\frac{1}{n}\cos x{\sin }^{n-1}x+\frac{n-1}{n}\int {\sin }^{n-2}xdx$$$$\int {\cos }^{n}xdx=\frac{1}{n}\sin x{\cos }^{n-1}x+\frac{n-1}{n}\int {\cos }^{n-2}xdx$$$$\int\mathrm{tan}^{{n}} \:{x}\:{dx}=\frac{\mathrm{1}}{{n}−\mathrm{1}}\mathrm{tan}^{{n}−\mathrm{1}}…

Question Number 129564 by bramlexs22 last updated on 16/Jan/21 $$\mathcal{V}=\int \frac{\sin \mathrm{x}}{\sin (\mathrm{x}+\theta )}\mathrm{dx}$$ Answered by Lordose last updated on 16/Jan/21 $$\mathrm{\Omega }=\int \frac{\sin \left(\mathrm{x}\right)}{\sin (\mathrm{x}+\theta )}\mathrm{dx}\stackrel{\mathrm{u}=\mathrm{x}+\theta }{=}\int \frac{\sin (\mathrm{u}-\theta )}{\sin \left(\mathrm{u}\right)}\mathrm{du}$$$$\mathrm{\Omega }=\int \frac{\sin \left(\mathrm{u}\right)\cos \theta -\cos \left(\mathrm{u}\right)\sin \theta }{\sin \left(\mathrm{u}\right)}\mathrm{du}=\mathrm{ucos}\theta -\sin \theta \ln \left(\mathrm{sinu}\right)+\mathrm{C}$$$$\Omega\:=\:\left(\mathrm{x}+\theta\right)\mathrm{cos}\theta\:−\:\mathrm{sin}\theta\mathrm{ln}\left(\mathrm{sin}\left(\mathrm{x}+\theta\right)\right)+\:\mathrm{C}…

Question Number 129519 by BHOOPENDRA last updated on 16/Jan/21 $$\int cos\left(y^3\right)dy$$ Answered by Dwaipayan Shikari last updated on 16/Jan/21 $$\int cos\left(y^3\right)dy$$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{e}^{{iy}^{\mathrm{3}}…