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Category: Integration

0-x-cos-t-sin-3-t-1-4-sin-x-sin-t-3-4-dt-

Question Number 130486 by benjo_mathlover last updated on 26/Jan/21 $$\:\int_{\mathrm{0}} ^{\:{x}} \:\frac{\mathrm{cos}\:{t}\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{3}} \:{t}}}{\left(\mathrm{sin}\:{x}−\mathrm{sin}\:{t}\right)^{\mathrm{3}/\mathrm{4}} }\:{dt}\:? \\ $$ Answered by MJS_new last updated on 26/Jan/21 $$\int\mathrm{cos}\:{t}\:\left(\frac{\mathrm{sin}\:{t}}{\mathrm{sin}\:{x}\:−\mathrm{sin}\:{t}}\right)^{\mathrm{3}/\mathrm{4}} {dt}=…

Integrate-the-function-f-x-y-xy-x-2-y-2-over-the-domain-R-3-x-2-y-2-3-1-y-4-

Question Number 130478 by liberty last updated on 26/Jan/21 $$\:{Integrate}\:{the}\:{function}\:{f}\left({x},{y}\right)={xy}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$${over}\:{the}\:{domain}\:{R}=\left\{−\mathrm{3}\leqslant{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \leqslant\mathrm{3},\:\mathrm{1}\leqslant{y}\leqslant\mathrm{4}\:\right\} \\ $$ Answered by EDWIN88 last updated on 26/Jan/21…

Given-that-f-x-f-pi-x-prove-that-0-pi-xf-x-dx-pi-2-0-pi-f-x-dx-please-what-are-different-methods-to-approach-this-question-

Question Number 130448 by physicstutes last updated on 25/Jan/21 $$\mathrm{Given}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{f}\left(\pi−{x}\right),\:\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx}\:=\:\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}. \\ $$$$\mathrm{please}\:\mathrm{what}\:\mathrm{are}\:\mathrm{different}\:\mathrm{methods}\:\mathrm{to}\:\mathrm{approach}\:\mathrm{this}\:\mathrm{question}? \\ $$ Answered by TheSupreme last updated on 25/Jan/21…

nice-calculus-prove-that-0-e-x-ln-x-cos-x-dx-1-8-4-pi-2ln-2-

Question Number 130433 by mnjuly1970 last updated on 25/Jan/21 $$\:\:\:\:\:\:….{nice}\:\:\:{calculus}… \\ $$$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}} {ln}\left({x}\right){cos}\left({x}\right){dx}\overset{?} {=}\frac{\mathrm{1}}{\mathrm{8}}\left(−\mathrm{4}\gamma−\pi−\mathrm{2}{ln}\left(\mathrm{2}\right)\right) \\ $$$$ \\ $$ Answered by Dwaipayan…