Question Number 91479 by Zainal Arifin last updated on 01/May/20 $$\:\underset{−\pi/\mathrm{2}\:} {\overset{\pi/\mathrm{2}} {\int}}\:\sqrt{\mathrm{cos}\:{x}−\mathrm{cos}^{\mathrm{3}} {x}}\:\mathrm{dx}=… \\ $$ Commented by jagoll last updated on 01/May/20 $$\sqrt{\mathrm{cos}\:{x}\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}\right)}\:=\:\sqrt{\mathrm{cos}\:{x}}\:\sqrt{\mathrm{sin}\:^{\mathrm{2}}…
Question Number 25932 by abdo imad last updated on 16/Dec/17 $${answer}\:{to}\:\mathrm{25824}\:\:{we}\:{have}\:{a}^{−{x}^{\mathrm{2}} } \:=\:{e}^{−{x}^{\mathrm{2}_{} } {ln}\left({a}\right)} \:\:{so}\:{for}\:{a}>\mathrm{1} \\ $$$${ln}\left({a}\right)=\left(\:\left({ln}\left({a}\right)\right)^{\mathrm{1}/\mathrm{2}} \right)^{\mathrm{2}} >>>>\int_{{R}} {a}^{−^{} {x}^{\mathrm{2}} } \:=\:\int_{{R}} {e}^{−\left({x}\:\left({ln}\left({a}\right)^{\mathrm{1}/\mathrm{2}}…
Question Number 25887 by shivram198922@gmail.com last updated on 16/Dec/17 Answered by ajfour last updated on 16/Dec/17 $$\int_{\mathrm{0}} ^{\:\:\pi} \left[\mathrm{cos}\:\left({p}−{n}\right){x}−\mathrm{cos}\:\left({p}+{n}\right){x}\right]{dx} \\ $$$$=\frac{\mathrm{sin}\:\left({p}−{n}\right)\pi}{{p}−{n}}−\frac{\mathrm{sin}\:\left({p}+{n}\right)\pi}{{p}+{n}} \\ $$$$=\frac{{p}\left[\mathrm{sin}\:\left({p}−{n}\right)\pi−\mathrm{sin}\:\left({p}+{n}\right)\pi\right]}{{p}^{\mathrm{2}} −{n}^{\mathrm{2}} }…
Question Number 25868 by ajfour last updated on 16/Dec/17 Commented by ajfour last updated on 16/Dec/17 $${Find}\:{the}\:{area}\:{enclosed}\:{by}\:{a} \\ $$$${variable}\:{line}\:\frac{{x}}{{t}}+\frac{{y}}{{l}−{t}}=\mathrm{1}\:{and} \\ $$$${the}\:{coordinate}\:{axes}\:{in}\:{the} \\ $$$${first}\:{quadrant};\: \\ $$$$\left({the}\:{line}\:{shifts}\:{as}\:{t}\:{changes}\right.…
Question Number 25865 by anonymous_ last updated on 15/Dec/17 $$\int\frac{\mathrm{2}+\mathrm{2}{x}}{\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$ Answered by ajfour last updated on 16/Dec/17 $$\frac{\mathrm{2}+\mathrm{2}{x}}{\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{{A}}{{x}−\mathrm{1}}+\frac{{Bx}+{C}}{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\Rightarrow\mathrm{2}+\mathrm{2}{x}={A}\left({x}^{\mathrm{2}}…
Question Number 25851 by abdo imad last updated on 15/Dec/17 $${find}\:{the}\:{value}\:{of}\:{integral}\:\:\:\int_{{R}} \left({z}−{a}\right)^{−\mathrm{1}} {dz}\:\:{with}\:{a}\:{from}\:{C}\:\:{aplly} \\ $$$${this}\:{result}\:{to}\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\infty} \left(\mathrm{2}\:+{x}^{\mathrm{4}_{} } \right)^{−\mathrm{1}} {dx}. \\ $$ Commented by abdo…
Question Number 156914 by cortano last updated on 17/Oct/21 Answered by puissant last updated on 17/Oct/21 $${D}=\int\frac{{arcsin}\left(\frac{\mathrm{1}}{{x}}\right)}{{x}^{\mathrm{5}} }{dx}\:;\:{u}=\frac{\mathrm{1}}{{x}}\:\rightarrow\:{du}=−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{dx} \\ $$$$\Rightarrow\:{D}=−\int\:\frac{{u}^{\mathrm{5}} {arcsin}\left({u}\right)}{{u}^{\mathrm{2}} }{du}=−\int{u}^{\mathrm{3}} {arcsin}\left({u}\right){du} \\…
Question Number 156915 by mnjuly1970 last updated on 17/Oct/21 Answered by qaz last updated on 07/Nov/21 $$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\left(\pi^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \right)\mathrm{cosh}\:\mathrm{x}} \\ $$$$=\int_{−\infty} ^{+\infty} \frac{\mathrm{e}^{\mathrm{x}}…
Question Number 25837 by mubeen897@hotmail.com last updated on 15/Dec/17 $$\int\left({x}^{{n}} {lnx}\right){dx} \\ $$ Answered by kaivan.ahmadi last updated on 15/Dec/17 $$\mathrm{u}=\mathrm{lnx}\Rightarrow\mathrm{du}=\frac{\mathrm{dx}}{\mathrm{x}} \\ $$$$\mathrm{dv}=\mathrm{x}^{\mathrm{n}} \mathrm{dx}\Rightarrow\mathrm{v}=\frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{n}+\mathrm{1}}…
Question Number 25824 by abdo imad last updated on 15/Dec/17 $${if}\:\:\int_{{R}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:\:=\:\:\pi^{\mathrm{1}/\mathrm{2}} \:\:\:\:{find}\:{value}\:\:{of}\:\:\:\int_{{R}} \:{a}^{−{x}^{\mathrm{2}_{} } } {dx}\:\:\:{with}\:{a}>\mathrm{0}\:{and}\:{a}\:{not}\:\mathrm{1}. \\ $$ Terms of Service Privacy…