Category: Integration

Question Number 201646 by Calculusboy last updated on 10/Dec/23 Answered by aleks041103 last updated on 10/Dec/23 $$\sqrt[ln\left(x\right)]{x}=x^{\frac{1}{ln\left(x\right)}}={\left(e^{ln\left(x\right)}\right)}^{\frac{1}{ln\left(x\right)}}=e=const$$$$\Rightarrow \int \sqrt[ln\left(x\right)]{x}dx=ex+C$$ Terms…

Question Number 201553 by Calculusboy last updated on 08/Dec/23 Answered by som(math1967) last updated on 09/Dec/23 $$1.\int \frac{1+logx-1}{{(1+logx)}^2}dx$$$$=\int \frac{dx}{(1+logx)}-\int \frac{dx}{{(1+logx)}^2}$$$$=\frac{1}{1+logx}\int dx-\int \{\frac{d}{dx}\times \frac{1}{1+logx}\int dx\}dx$$$$\:\:\:\:\:\:−\int\frac{{dx}}{\left(\mathrm{1}+{logx}\right)^{\mathrm{2}}…

Question Number 201546 by Calculusboy last updated on 08/Dec/23 $$\int \mathit{S}\mathit{i}\mathit{n}\left(\mathit{I}\mathit{n}\mathit{x}\right)\mathit{d}\mathit{x}$$ Commented by aleks041103 last updated on 09/Dec/23 $$isthissineofnaturallogofxorsthelse?$$ Commented by Calculusboy…

Question Number 201509 by Calculusboy last updated on 07/Dec/23 Answered by witcher3 last updated on 09/Dec/23 $$\mathrm{tack}\mathrm{principal}\mathrm{definition}\mathrm{of}\mathrm{Log}\left(\mathrm{z}\right)=\ln \mid \mathrm{z}\mid +\mathrm{iarg}\left(\mathrm{z}\right)$$$$\mathrm{z}\in ]-\frac{\pi }{2},\frac{3\pi }{2}[$$$$\ln (\mathrm{ix}+\ln \left(\cos \left(\mathrm{x}\right)\right)=\ln \mid \sqrt{{\mathrm{x}}^2+{\ln }^2(\cos \left(\mathrm{x}\right)}\mid +\mathrm{i}\arg (\ln (\cos \left(\mathrm{x}\right)+\mathrm{ix})$$$$\mathrm{arg}\left(\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{x}\right)+\mathrm{ix}\right)\in\left[\frac{\pi}{\mathrm{2}},\pi\left[\:\right.\right.\right.…

Question Number 201452 by tri26112004 last updated on 06/Dec/23 Answered by Calculusboy last updated on 06/Dec/23 $$\int {\mathit{x}}^{-2}{\mathit{e}}^{-4\mathit{x}}\mathit{d}\mathit{x}$$$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:\mathit{b}\mathit{y}\mathit{u}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{I}\mathit{B}\mathit{P}$$$$\boldsymbol{{let}}\:\boldsymbol{{u}}=\boldsymbol{{e}}^{−\mathrm{4}\boldsymbol{{x}}} \:\:\:\boldsymbol{{du}}=−\mathrm{4}\boldsymbol{{e}}^{−\mathrm{4}\boldsymbol{{x}}} \boldsymbol{{dx}}\:\:\boldsymbol{{dv}}=\boldsymbol{{x}}^{−\mathrm{2}}…