Category: Integration

Question Number 213628 by Thomaseinstein last updated on 10/Nov/24 Answered by MrGaster last updated on 03/Feb/25 $$=\frac{1}{2}\left(x^4\sin \left({\tan }^{-1}\left(2x\right)\right)\right)-\frac{1}{2}{x}^3\cos \left({\tan }^{-1}\left(2x\right)\right))$$ Terms of…

Question Number 213342 by Spillover last updated on 03/Nov/24 Commented by MrGaster last updated on 03/Nov/24 $$\overline{)\begin{array}{c}\int H_x^{\sqrt{\pi }}dx=\frac{H_x^{\sqrt{\pi }+1}}{\sqrt{\pi }+1}+C\end{array}}$$ Terms of…

Question Number 213376 by RoseAli last updated on 03/Nov/24 $$\int \frac{dx}{\sqrt{{(4x^2+1)}^3}}$$ Commented by Frix last updated on 03/Nov/24 $$\mathrm{Sometimes}\mathrm{just}\mathrm{use}\mathrm{your}\mathrm{brain}\mathrm{\&}\mathrm{experience}$$$$\frac{{d}}{{dx}}\left[\frac{{g}\left({x}\right)}{\:\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}}}\right]=\frac{{g}'\left({x}\right)\left(\mathrm{4}{x}^{\mathrm{2}}…

Question Number 213216 by Spillover last updated on 01/Nov/24 Answered by MrGaster last updated on 01/Nov/24 $$=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} }{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{3}} +\mathrm{1}\right)}{dx}+\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{log}^{\mathrm{2}}…

Question Number 213138 by Spillover last updated on 31/Oct/24 Commented by Frix last updated on 31/Oct/24 $$\mathrm{We}\mathrm{must}\mathrm{first}\mathrm{solve}\int \sin t^{3}dt\mathrm{which}\mathrm{might}$$$$\mathrm{be}\mathrm{possible}\mathrm{using}$$$$\mathrm{sin}\:{t}^{\mathrm{3}} \:=\frac{\mathrm{e}^{\mathrm{i}{t}^{\mathrm{3}} } −\mathrm{e}^{−\mathrm{i}{t}^{\mathrm{3}}…