Category: Integration

Question Number 116586 by Bird last updated on 05/Oct/20 $$1)explicite{\int }_0^{\mathrm{\infty }}\frac{arctan(1+x(2+t^2))}{2+t^2}dt$$$$withx>0$$$$2)determinevaluesof$$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{3}+{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt}\:{and}\: \…

Question Number 116564 by bemath last updated on 05/Oct/20 $$\int \frac{\mathrm{dx}}{\mathrm{x}+\sqrt[3]{\mathrm{x}}}?$$ Answered by MJS_new last updated on 05/Oct/20 $$\int \frac{dx}{x^{1/3}(x^{2/3}+1)}=$$$$\:\:\:\:\:\left[{t}={x}^{\mathrm{2}/\mathrm{3}} +\mathrm{1}\:\rightarrow\:{dx}=\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{1}/\mathrm{3}}…

Question Number 116557 by Bird last updated on 04/Oct/20 $$calculate{\int }_0^{\mathrm{\infty }}\frac{ln(1+x(1+t^2)}{1+t^2}dt$$$$withx>0$$$$2)findthevalueof{\int }_0^{\mathrm{\infty }}\frac{ln(2+t^2)}{1+t^2}dt$$$${and}\:\int_{\mathrm{0}} ^{\infty}…

Question Number 116533 by Bird last updated on 04/Oct/20 $$find{u}_n={\int }_0^{\mathrm{\infty }}\frac{{sin}^{n}x}{x}dx(n⩾1)$$ Answered by Olaf last updated on 04/Oct/20 $$\mathrm{1st}\:\mathrm{case}\::\:{n}\:\mathrm{is}\:\mathrm{odd} \…

Question Number 116501 by bemath last updated on 04/Oct/20 $$\int \frac{\sin \mathrm{x}-\cos \mathrm{x}}{\sqrt{\sin 2\mathrm{x}}}\mathrm{dx}?$$ Answered by TANMAY PANACEA last updated on 04/Oct/20 $$\frac{d}{dx}(sinx+cosx)=cosx-sinx$$$$\int \frac{-d(sinx+cosx)}{\sqrt{1-1+sin2x}}$$$$\int\frac{−{d}\left({sinx}+{cosx}\right)}{\:\sqrt{\left({sinx}+{cosx}\right)^{\mathrm{2}}…