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Category: Integration

cos-2x-1-sin-2-x-dx-

Question Number 138691 by liberty last updated on 16/Apr/21 $$\int\:\mathrm{cos}\:\mathrm{2}{x}\:\sqrt{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} {x}}\:{dx}\:=? \\ $$ Answered by mathmax by abdo last updated on 17/Apr/21 $$\mathrm{I}=\int\:\mathrm{cos}\left(\mathrm{2x}\right)\sqrt{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}}\mathrm{dx}\:\Rightarrow\mathrm{I}=\int\:\mathrm{cos}\left(\mathrm{2x}\right)\sqrt{\mathrm{1}+\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{2}}}\mathrm{dx} \\…

nice-calculus-find-the-value-of-n-1-1-n-sin-2-n-n-

Question Number 138683 by mnjuly1970 last updated on 16/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:..\:…\:…\:{calculus}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Theta=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {sin}^{\mathrm{2}} \left({n}\right)}{{n}}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……………………. \\ $$ Answered by Dwaipayan…

I-dx-px-q-ax-2-bx-c-

Question Number 138656 by ajfour last updated on 16/Apr/21 $${I}=\int\frac{{dx}}{\left({px}+{q}\right)\sqrt{{ax}^{\mathrm{2}} +{bx}+{c}}} \\ $$ Answered by Ar Brandon last updated on 16/Apr/21 $$\mathcal{I}=\int\frac{\mathrm{dx}}{\left(\mathrm{px}+\mathrm{q}\right)\sqrt{\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}}} \\ $$$$\mathrm{u}=\frac{\mathrm{1}}{\mathrm{px}+\mathrm{q}}\:\Rightarrow\mathrm{x}=\frac{\mathrm{1}}{\mathrm{up}}−\frac{\mathrm{q}}{\mathrm{p}}\Rightarrow\mathrm{du}=−\frac{\mathrm{p}}{\left(\mathrm{px}+\mathrm{q}\right)^{\mathrm{2}}…

0-1-x-1-x-2-dx-

Question Number 7585 by Tawakalitu. last updated on 04/Sep/16 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:{dx} \\ $$ Commented by sou1618 last updated on 05/Sep/16 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}}…

Question-138621

Question Number 138621 by mnjuly1970 last updated on 15/Apr/21 Answered by Dwaipayan Shikari last updated on 15/Apr/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\Gamma^{\mathrm{2}} \left({n}+\mathrm{1}\right)}{\Gamma\left(\mathrm{2}{n}+\mathrm{2}\right)}=\int_{\mathrm{0}} ^{\mathrm{1}} \Sigma{x}^{{n}} \left(\mathrm{1}−{x}\right)^{{n}} {dx}…

Given-a-function-f-where-f-x-0for-x-R-If-the-area-U-x-y-0-2y-f-x-6-x-2-is-u-and-the-area-V-x-y-0-y-f-x-2-x-0-is-v-then-what-the-value-of-1-2-4x-f-2x-2-8-dx-A-5u-4v-

Question Number 138611 by liberty last updated on 15/Apr/21 $${Given}\:{a}\:{function}\:{f}\:{where}\: \\ $$$${f}\left({x}\right)\geqslant\:\mathrm{0}{for}\:\forall{x}\in\mathbb{R}.\:{If}\:{the}\:{area} \\ $$$${U}\:=\:\left\{\:\left({x},{y}\right)\mid\mathrm{0}\leqslant\mathrm{2}{y}\leqslant{f}\left({x}\right),\:−\mathrm{6}\leqslant{x}\leqslant−\mathrm{2}\right\} \\ $$$${is}\:{u}\:{and}\:{the}\:{area}\:{V}=\left\{\left({x},{y}\right)\mid\mathrm{0}\leqslant{y}\leqslant{f}\left({x}\right),−\mathrm{2}\leqslant{x}\leqslant\mathrm{0}\right\} \\ $$$${is}\:{v}\:{then}\:{what}\:{the}\:{value}\:{of} \\ $$$$\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\mathrm{4}{x}\:{f}\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{8}\right)\:{dx}\:. \\ $$$$\left({A}\right)\:\mathrm{5}{u}+\mathrm{4}{v}\:\:\:\:\:\left({D}\right)\mathrm{2}{u}+{v}…