Category: Integration

Question Number 129173 by bramlexs22 last updated on 13/Jan/21 $$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\tan \theta +\cot \theta +\sec \theta +\csc \theta }?$$ Answered by liberty last updated on 13/Jan/21 $$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\frac{\sin \theta +1}{\cos \theta }+\frac{\cos \theta +1}{\sin \theta }}=\int \frac{\cos \theta \sin \theta \mathrm{d}\theta }{\sin {}^2\theta +\sin \theta +\cos {}^2\theta +\cos \theta }$$$$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{d}\theta}{\mathrm{1}+\mathrm{sin}\:\theta+\mathrm{cos}\:\theta}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\theta}{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}\mathrm{d}\theta…

Question Number 129159 by gowsalya last updated on 13/Jan/21 Answered by TheSupreme last updated on 13/Jan/21 $$\int (e^{i\theta }+e^{-i\theta })e^{i\theta }/2=\frac{e^{i\theta }}{4i}+\frac{\theta }{2}+c$$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} …=\pi…

Question Number 129150 by benjo_mathlover last updated on 13/Jan/21 $${\int }_0^{\mathrm{x}}{(\mathrm{x}-\mathrm{u})}^2\mathrm{u}\mathrm{f}\left(\mathrm{u}\right)\mathrm{du}=4\mathrm{x}-6\sin \mathrm{x}+2\mathrm{xcos}\mathrm{x}$$$$\mathrm{f}\left(\mathrm{x}\right)=?$$ Answered by liberty last updated on 13/Jan/21 $$\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{0}}…

Question Number 63566 by aliesam last updated on 05/Jul/19 $$provethat$$$$$$$$\int {sin}^n\left(x\right)dx,p\in n,p⩾2=-\frac{1}{n}cos\left(x\right){sin}^{n-1}\left(x\right)+(p-1)\int {sin}^{n-2}\left(x\right)dx$$ Commented by aliesam last updated on…

Question Number 129064 by Eric002 last updated on 12/Jan/21 $$valuatethefollowingintegral$$$$I={\int }_1^{\mathrm{\infty }}\frac{dt}{{\left(t\right)}^{2k+v+\frac{3}{2}}\sqrt{t-1}}$$$$andprovethat:$$$$I=\sqrt{\pi }\frac{\mathrm{\Gamma }(v+1)}{\mathrm{\Gamma }(v+\frac{3}{2})}\left(\frac{{\left(\frac{v+1}{2}\right)}_k{(1+\frac{v}{2})}_k}{{(\frac{v}{2}+\frac{3}{4})}_k{(\frac{v}{2}+\frac{5}{4})}_k}\right)$$…