Category: Integration

Question Number 196557 by BHOOPENDRA last updated on 27/Aug/23 Answered by qaz last updated on 27/Aug/23 $$\int_{\mathrm{0}} ^{{t}} {e}^{−{u}} \mathrm{sin}\:{udu}=−\Im\int_{\mathrm{0}} ^{{t}} {e}^{−\left(\mathrm{1}+{i}\right){u}} {du}=\Im\frac{\mathrm{1}}{\mathrm{1}+{i}}\left({e}^{−\left(\mathrm{1}+{i}\right){t}} −\mathrm{1}\right) \…

Question Number 196459 by RoseAli last updated on 25/Aug/23 Answered by Frix last updated on 25/Aug/23 $$\mathrm{Use}{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\mathrm{to}\mathrm{get}$$$$\int\frac{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }{dx}=\frac{{x}\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}\right)}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dx}}{{x}^{\mathrm{2}}…

Question Number 196436 by RoseAli last updated on 24/Aug/23 $$\int \frac{dx}{x(x^n-1)}$$ Answered by MM42 last updated on 24/Aug/23 $$\int\frac{{x}^{{n}} −\mathrm{1}+{x}^{{n}} }{{x}\left({x}^{{n}} −\mathrm{1}\right)}=\int\left(\frac{\mathrm{1}}{{x}}+\frac{{x}^{{n}−\mathrm{1}} }{{x}^{{n}}…

Question Number 196375 by RoseAli last updated on 23/Aug/23 $$\int \frac{dx}{x(x^4-1)}$$ Commented by RoseAli last updated on 23/Aug/23 $$\int \frac{dx}{x(x^4+1)}$$ Answered…

Question Number 196277 by cortano12 last updated on 21/Aug/23 $$\cancel{\underset{―}{\underbrace{}}}$$ Answered by mr W last updated on 21/Aug/23 $$sayz={\sin }^{3}x$$$$\frac{{dz}}{{dy}}=\frac{{dz}}{{dx}}×\frac{{dx}}{{dy}}=\mathrm{3}\:\mathrm{sin}^{\mathrm{2}}…