Question Number 214770 by MathematicalUser2357 last updated on 19/Dec/24 $$\int\mathrm{sin}^{\mathrm{3}} {xdx}=? \\ $$ Answered by mr W last updated on 19/Dec/24 $$=−\int\mathrm{sin}^{\mathrm{2}} \:{x}\:{d}\:\left(\mathrm{cos}\:{x}\right) \\ $$$$=−\int\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}}…
Question Number 214750 by universe last updated on 18/Dec/24 Answered by aleks041103 last updated on 21/Dec/24 $${J}=\int_{\mathrm{0}} ^{\infty} \int_{{z}} ^{\infty} \int_{{z}} ^{\infty} {cos}\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right){dxdydz}…
Question Number 214756 by MathematicalUser2357 last updated on 19/Dec/24 $$\int\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}\mathrm{sin}\:\sqrt{{x}}{dx}=? \\ $$ Answered by mr W last updated on 19/Dec/24 $$=\int\mathrm{sin}\:\sqrt{{x}}\:{d}\left(\sqrt{{x}}\right) \\ $$$$=−\mathrm{cos}\:\sqrt{{x}}+{C} \\ $$…
Question Number 214661 by efronzo1 last updated on 15/Dec/24 Answered by Rasheed.Sindhi last updated on 15/Dec/24 $${x}.{f}\left({x}\right)+{x}.{f}\left(\frac{\mathrm{1}}{{x}}\right)={x}+\mathrm{1} \\ $$$${f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)=\frac{{x}+\mathrm{1}}{{x}}……{A} \\ $$$${Replace}\:{x}\:{by}\:\:\frac{\mathrm{1}}{{x}}: \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)+\:{f}\left({x}\right)=\frac{\frac{\mathrm{1}}{{x}}+\mathrm{1}}{\frac{\mathrm{1}}{{x}}}={x}+\mathrm{1}…{B} \\ $$$${A}\:\&\:{B}:\:\:\:\frac{{x}+\mathrm{1}}{{x}}={x}+\mathrm{1}…
Question Number 214618 by kuldeep52 last updated on 13/Dec/24 $$\int_{\mathrm{0}} ^{\Pi/\mathrm{2}} \frac{\mathrm{3}\sqrt{\mathrm{tan}\:{x}}}{\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} \:}{dx} \\ $$ Answered by Frix last updated on 14/Dec/24 $$\mathrm{3}\int\frac{\sqrt{\mathrm{tan}\:{x}}}{\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)^{\mathrm{2}} }{dx}\:\overset{\left[{t}=\sqrt{\mathrm{tan}\:{x}}\right]} {=}…
Question Number 214568 by Ikbal last updated on 12/Dec/24 $$\int\frac{{b}+{ax}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx} \\ $$ Answered by MathematicalUser2357 last updated on 18/Dec/24 $$\int\frac{{ax}+{b}}{\mathrm{sin}\:{x}+\mathrm{1}}{dx} \\ $$$$=\int\frac{\left({ax}+{b}\right)\mathrm{sec}\:{x}}{\mathrm{tan}\:{x}+\mathrm{sec}\:{x}}{dx} \\ $$$$=−\frac{{ax}+{b}}{\mathrm{tan}\:{x}+\mathrm{sec}\:{x}}+\int\frac{{a}}{\mathrm{tan}\:{x}+\mathrm{sec}\:{x}}{dx} \\…
Question Number 214560 by ajfour last updated on 12/Dec/24 $$\int_{\mathrm{1}} ^{\:\:{x}} \frac{\mathrm{ln}\:{x}}{\:\sqrt{\mathrm{1}−\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} }}{dx} \\ $$ Answered by MathematicalUser2357 last updated on 18/Dec/24 $$\:\cancel{\lesseqgtr} \\ $$…
Question Number 214509 by depressiveshrek last updated on 10/Dec/24 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{of}\:\mathrm{revolution} \\ $$$$\mathrm{generated}\:\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded} \\ $$$$\mathrm{by}\:{y}={x}\left(\mathrm{2}−{x}\right)\:\mathrm{and}\:{y}={x}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}−\mathrm{axis}. \\ $$ Commented by mr W last updated on 11/Dec/24 $${you}\:{seem}\:{never}\:{to}\:{give}\:{any}\:{feedback}…
Question Number 214498 by universe last updated on 10/Dec/24 Answered by mr W last updated on 10/Dec/24 $$=\mathrm{4}\pi\int_{\mathrm{0}} ^{\infty} {re}^{−{r}^{\mathrm{2}} } {r}^{\mathrm{2}} {dr} \\ $$$$=\mathrm{2}\pi\int_{\mathrm{0}}…
Question Number 214360 by shunmisaki007 last updated on 06/Dec/24 $$\mathrm{Find}\:\underset{−\infty} {\overset{\infty} {\int}}{e}^{−{ax}^{\mathrm{2}} } {dx}\:\mathrm{when}\:{a}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{without}\:\mathrm{changing}\:\mathrm{the}\:\mathrm{coordinate}. \\ $$ Answered by mathmax last updated on 06/Dec/24 $$=\mathrm{2}\int_{\mathrm{0}} ^{\infty}…