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Category: Integration

Prove-0-pi-2-d-0-pi-2-f-sin-cos-sin-d-pi-2-0-1-f-x-dx-

Question Number 216281 by MrGaster last updated on 02/Feb/25 $$\mathrm{Prove}:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {d}\phi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\theta\:\mathrm{cos}\:\theta\right)\mathrm{sin}\theta\:{d}\theta=\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$ Commented by mr W last updated on…

Question-216093

Question Number 216093 by mnjuly1970 last updated on 27/Jan/25 Answered by mahdipoor last updated on 27/Jan/25 $${note}\::\:\mathrm{0}\leqslant{x}<\mathrm{1}\:\:,\:\:{y}^{\mathrm{2}} =\left(\mathrm{1}+{y}^{\mathrm{2}} \right){x} \\ $$$${get}\:\:{f}={ln}\left({y}\right)+{ln}\left({y}'\right) \\ $$$$\Rightarrow\frac{{df}}{{dx}}=\frac{{y}^{'} }{{y}}+\frac{{y}^{''} }{{y}'}=\mathrm{2}\varphi…

i-sec-5-d-ii-tan-d-cos-

Question Number 216042 by ajfour last updated on 26/Jan/25 $$\left({i}\right)\:\:\:\int\mathrm{sec}\:^{\mathrm{5}} \theta{d}\theta \\ $$$$\left({ii}\right)\:\:\int\:\frac{\sqrt{\mathrm{tan}\:\theta}\:{d}\theta}{\mathrm{cos}\:\theta} \\ $$ Commented by ajfour last updated on 26/Jan/25 https://youtu.be/VqEdd_VGxGI?si=VnVRn7bW5wENyi9Y Find radius of circle inscribed in half ellipse. Commented by…

Question-215963

Question Number 215963 by RoseAli last updated on 23/Jan/25 Answered by Frix last updated on 23/Jan/25 $$\int\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)^{{n}} \left(\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:{x}\right){dx}= \\ $$$$=\int\left(\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\right)^{{n}} \left(\mathrm{sin}^{\mathrm{2}} \:{x}\:−\mathrm{cos}^{\mathrm{2}} \:{x}\right){dx}= \\…

2sin-1-x-x-1-x-2-2-x-4-dx-

Question Number 215974 by ajfour last updated on 23/Jan/25 $$\int\sqrt{\left(\mathrm{2sin}^{−\mathrm{1}} {x}−{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right)^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$ Answered by MathematicalUser2357 last updated on 26/Jan/25 $$\mathrm{No}\:\mathrm{closed}\:\mathrm{forms}.\:\frac{{x}\mid{x}\mid}{\mathrm{2}}+\frac{{x}^{\mathrm{3}} \mid{x}\mid}{\mathrm{3}}−\frac{\mathrm{2}{x}^{\mathrm{5}}…