Question Number 20389 by tammi last updated on 26/Aug/17 $$\int\mathrm{sin}\:^{\mathrm{4}} {xdx} \\ $$ Answered by Joel577 last updated on 26/Aug/17 $${I}\:=\:\int\:\left(\frac{\mathrm{1}\:−\:\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2}}\right)^{\mathrm{2}} \:{dx} \\ $$$$\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:\int\:\left(\mathrm{1}\:−\:\mathrm{2cos}\:\mathrm{2}{x}\:+\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{2}{x}\right)\:{dx}…
Question Number 20390 by tammi last updated on 26/Aug/17 $$\int\mathrm{sin}\:^{\mathrm{5}} {xdx} \\ $$ Answered by ajfour last updated on 26/Aug/17 $$\int\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}\right)^{\mathrm{2}} \mathrm{sin}\:{x}\:{dx} \\ $$$${let}\:\mathrm{cos}\:{x}={t}\:\:\Rightarrow\:\:\:−\mathrm{sin}\:{xdx}={dt}…
Question Number 20387 by tammi last updated on 26/Aug/17 $$\int\mathrm{sin}\:{px}\mathrm{cos}\:{qxdx} \\ $$ Answered by mrW1 last updated on 26/Aug/17 $$\mathrm{sin}\:\left(\mathrm{p}+\mathrm{q}\right)\mathrm{x}=\mathrm{sin}\:\mathrm{px}\:\mathrm{cos}\:\mathrm{qx}+\mathrm{cos}\:\mathrm{px}\:\mathrm{sin}\:\mathrm{qx} \\ $$$$\mathrm{sin}\:\left(\mathrm{p}−\mathrm{q}\right)\mathrm{x}=\mathrm{sin}\:\mathrm{px}\:\mathrm{cos}\:\mathrm{qx}−\mathrm{cos}\:\mathrm{px}\:\mathrm{sin}\:\mathrm{qx} \\ $$$$\Rightarrow\mathrm{sin}\:\mathrm{px}\:\mathrm{cos}\:\mathrm{qx}=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{sin}\:\left(\mathrm{p}+\mathrm{q}\right)\mathrm{x}+\mathrm{sin}\:\left(\mathrm{p}−\mathrm{q}\right)\mathrm{x}\right] \\…
Question Number 20388 by tammi last updated on 26/Aug/17 $$\int\frac{\mathrm{2sin}\:{x}+\mathrm{3cos}\:{x}}{\mathrm{7sin}\:{x}−\mathrm{2cos}\:{x}} \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${Let}\:\mathrm{2}{sinx}+\mathrm{3}{cosx}={A}\left(\mathrm{7}{sinx}−\mathrm{2}{cosx}\right)+{B}\left(\mathrm{7}{cosx}+\mathrm{2}{sinx}\right) \\ $$$$\Rightarrow\mathrm{7}{A}−\mathrm{2}{B}=\mathrm{2} \\ $$$$\:\:\:\:−\mathrm{2}{A}+\mathrm{7}{B}=\mathrm{3} \\…
Question Number 20384 by tammi last updated on 26/Aug/17 $$\int\frac{{x}^{\mathrm{3}} {dx}}{\left(\mathrm{2}+\mathrm{3}{x}\right)^{\mathrm{2}} } \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${let}\:\mathrm{2}+\mathrm{3}{x}={t}\:\Rightarrow\mathrm{3}{dx}={dt}\:\&\:{x}=\frac{{t}−\mathrm{2}}{\mathrm{3}} \\ $$$$\int\left(\frac{{t}−\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{3}} \frac{{dt}}{\mathrm{3}{t}^{\mathrm{2}}…
Question Number 20385 by tammi last updated on 26/Aug/17 $$\int\sqrt{\mathrm{16}−\mathrm{9}{x}^{\mathrm{2}} {dx}} \\ $$ Answered by ajfour last updated on 26/Aug/17 $$=\mathrm{3}\int\sqrt{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{2}} −{x}^{\mathrm{2}} }\:{dx} \\ $$$$=\frac{\mathrm{3}{x}}{\mathrm{2}}\sqrt{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{2}}…
Question Number 20386 by tammi last updated on 26/Aug/17 $$\int\sqrt{\mathrm{1}−{a}^{\mathrm{2}} {x}^{\mathrm{2}} {dx}} \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${Let}\:{x}=\frac{{sin}\theta}{{a}} \\ $$$$\Rightarrow{dx}=\frac{\mathrm{1}}{{a}}{cos}\theta{d}\theta \\…
Question Number 20383 by tammi last updated on 26/Aug/17 $$\int\frac{{dx}}{{x}\sqrt{\mathrm{2}+\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }} \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${let}\:\mathrm{2}+{x}^{\frac{\mathrm{1}}{\mathrm{3}}} ={t}^{\mathrm{2}} \Rightarrow{x}=\left({t}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{3}} \\…
Question Number 85914 by frc2crc last updated on 26/Mar/20 $${Let}\:{I}_{{n}} =\overset{\pi/\mathrm{4}} {\int}_{\mathrm{0}\:} \left(\frac{\mathrm{1}−\mathrm{tan}\:{A}}{\mathrm{1}+\mathrm{tan}\:{A}}\right)^{{n}} {dA}\:\:{what}\:{is} \\ $$$${the}\:{Laplace}\:{Transform}\:{and}\:{the} \\ $$$${Fourier}\:{Transform} \\ $$ Answered by mind is power…
Question Number 85909 by jagoll last updated on 26/Mar/20 $$\int\:\mathrm{sin}^{−\mathrm{1}} \:\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\right)\:\mathrm{dx}\:,\:\mathrm{a}\:>\:\mathrm{0} \\ $$ Commented by john santu last updated on 26/Mar/20 $${let}\:\sqrt{\frac{{x}}{{a}+{x}}}\:=\:{n}\:\Rightarrow\:{x}\:=\:\frac{{an}^{\mathrm{2}} }{\mathrm{1}−{n}^{\mathrm{2}} } \\…