Question Number 11621 by Nayon last updated on 29/Mar/17 $${Evaluate}\:\:\int\frac{{ax}+{b}}{{cx}+{d}}{dx} \\ $$ Answered by mrW1 last updated on 29/Mar/17 $$=\int\frac{\frac{{a}}{{c}}\left({cx}+{d}\right)+\left({b}−\frac{{ad}}{{c}}\right)}{{cx}+{d}}{dx} \\ $$$$=\frac{{a}}{{c}}\int{dx}+\left({b}−\frac{{ad}}{{c}}\right)×\frac{\mathrm{1}}{{c}}\int\frac{{d}\left({cx}+{d}\right)}{{cx}+{d}} \\ $$$$=\frac{{ax}}{{c}}+\frac{{bc}−{ad}}{{c}^{\mathrm{2}} }×\mathrm{ln}\:\left({cx}+{d}\right)+{C}…
Question Number 77158 by TawaTawa last updated on 03/Jan/20 $$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{3}} \:\mathrm{cos}\frac{\mathrm{x}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}\:−\:\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$ Commented by TawaTawa last updated on 03/Jan/20 $$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\…
Question Number 142689 by rs4089 last updated on 04/Jun/21 Answered by Dwaipayan Shikari last updated on 04/Jun/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}\right)}{{x}}{f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({x}\right){dx}\:\:\:\:{f}\left({x}\pm\pi\right)={f}\left({x}\right) \\ $$$${let}\:{f}\left({x}\right)={sin}^{\mathrm{2}{n}−\mathrm{2}} {x}\:\:\:\:\:\:…
Question Number 11619 by Nayon last updated on 29/Mar/17 $${Evaluate}\:\int\frac{{dx}}{\left({x}+{a}\right)\left({x}+{b}\right)} \\ $$ Commented by Nayon last updated on 29/Mar/17 $${someone}\:{pls}\:{ans} \\ $$ Answered by mrW1…
Question Number 142687 by iloveisrael last updated on 04/Jun/21 $$\:\:\:\:\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:=? \\ $$ Answered by qaz last updated on 04/Jun/21 $$\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}\sqrt{\mathrm{1}+\mathrm{cos}\:\mathrm{x}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int\frac{\mathrm{dx}}{\left(\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{2}}−\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int\frac{\mathrm{dx}}{\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}−\mathrm{cos}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}…
Question Number 11616 by Nayon last updated on 29/Mar/17 $${EvAluate}\:\int\left({x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{9}} {dx} \\ $$ Answered by Joel576 last updated on 29/Mar/17 $$\mathrm{Let}\:{u}\:=\:{x}^{\mathrm{2}} \:+\:\mathrm{9} \\ $$$$\frac{{du}}{{dx}}\:=\:\mathrm{2}{x}\:\:\Leftrightarrow\:\:{dx}\:=\:\frac{{du}}{\mathrm{2}{x}}\:\:…
Question Number 11609 by Nayon last updated on 29/Mar/17 $$ \\ $$$$ \\ $$$${Evaluate}\:\int{x}^{{x}} {dx} \\ $$ Commented by Nayon last updated on 29/Mar/17 $${pls}\:{ans}\:{mrw}\mathrm{1}…
Question Number 142667 by mnjuly1970 last updated on 03/Jun/21 Answered by mindispower last updated on 03/Jun/21 $${T}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {ln}\left(\mathrm{1}+{x}\right){dx}=\frac{{ln}\left(\mathrm{2}\right)}{{n}+\mathrm{1}}−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}.\frac{{dx}}{\mathrm{1}+{x}} \\ $$$$\mathrm{0}\leqslant\int_{\mathrm{0}}…
Question Number 142656 by qaz last updated on 03/Jun/21 $$\int_{−\pi} ^{\pi} \frac{\mathrm{xsin}\:\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=? \\ $$ Answered by Ar Brandon last updated on 03/Jun/21 $$\xi\left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\pi}…
Question Number 142643 by ArielVyny last updated on 03/Jun/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{cos}^{\mathrm{2}} {t}}{{sint}}{dt} \\ $$ Answered by MJS_new last updated on 03/Jun/21 $$\int\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\frac{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\left(−\mathrm{sin}\:{t}\:+\mathrm{csc}\:{t}\right){dt}=…