Question Number 187703 by Tawa11 last updated on 20/Feb/23 $$\int\:\frac{\mathrm{1}}{\mathrm{5x}^{\mathrm{2}} \:\:−\:\:\mathrm{2x}\:\:−\:\:\mathrm{4}}\:\mathrm{dx} \\ $$ Answered by MikeH last updated on 20/Feb/23 $$=\:\frac{\mathrm{1}}{\mathrm{5}}\int\frac{\mathrm{1}}{{x}^{\mathrm{2}} −\frac{\mathrm{2}}{\mathrm{5}}{x}−\frac{\mathrm{4}}{\mathrm{5}}}\:{dx} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{5}}\int\frac{\mathrm{1}}{\left({x}−\frac{\mathrm{1}}{\mathrm{5}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{25}}−\frac{\mathrm{4}}{\mathrm{5}}}{dx}…
Question Number 122160 by benjo_mathlover last updated on 14/Nov/20 Answered by liberty last updated on 14/Nov/20 $$\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\left[\:\mathrm{f}\left(\mathrm{x}\right)+\mathrm{1}\:\right]\:\mathrm{dx}\:−\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\left[\:\mathrm{f}\left(\mathrm{t}\right)+−\mathrm{1}\right]\:\mathrm{dt}\:= \\ $$$$\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}−\int_{\mathrm{0}}…
Question Number 122159 by mnjuly1970 last updated on 14/Nov/20 $$\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:{prove}\:\:{that}:: \\ $$$$\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left\{{tan}^{−\mathrm{1}} \left({ptan}\left({x}\right)\right)−{tan}^{−\mathrm{1}} \left({qtan}\left({x}\right)\right)\right\}\left({tan}\left({x}\right)+{cot}\left({x}\right)\right){dx} \\ $$$$=\frac{\pi}{\mathrm{2}}\:{log}\left(\frac{{p}}{{q}}\right)\:\:\:\left(\:\:\:\:{p}\:,\:{q}\:>\mathrm{0}\:\:\:\right) \\ $$$$\:\:\:\:{m}.{n}. \\ $$ Answered…
Question Number 122157 by mathmax by abdo last updated on 14/Nov/20 $$\mathrm{find}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$ Commented by peter frank last updated on 14/Nov/20…
Question Number 122137 by sdfg last updated on 14/Nov/20 Answered by Bird last updated on 14/Nov/20 $$\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\int_{−\infty} ^{+\infty} \:{e}^{{xt}−{e}^{{t}}…
Question Number 122120 by sina1377 last updated on 14/Nov/20 $$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}}{\:\sqrt{{y}}}.{e}^{{y}} {dy} \\ $$ Answered by Bird last updated on 14/Nov/20 $${I}\:=\int_{\mathrm{0}} ^{\mathrm{3}} \:\frac{{e}^{{y}}…
Question Number 122108 by bemath last updated on 14/Nov/20 $$\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{ln}\:\left({x}\right)}{{x}^{\mathrm{2}} }\:{dx}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 14/Nov/20 $$−\left[{log}\left({x}\right)\frac{\mathrm{1}}{{x}}\right]_{\mathrm{1}} ^{\mathrm{2}}…
Question Number 122109 by sdfg last updated on 14/Nov/20 Answered by Dwaipayan Shikari last updated on 14/Nov/20 $$\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$$\Gamma\left({x}\right)=\int_{\mathrm{1}} ^{\infty}…
Question Number 122098 by Rohit412 last updated on 14/Nov/20 Commented by liberty last updated on 14/Nov/20 $$\int\:\frac{\mathrm{dx}}{\:\sqrt{\left(\mathrm{2ax}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} }}\:=\:\int\:\frac{\mathrm{dx}}{\left(\sqrt{\mathrm{2ax}+\mathrm{x}^{\mathrm{2}} }\right)^{\mathrm{3}} } \\ $$$$\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{3}} \:\left(\sqrt{\mathrm{2ax}^{−\mathrm{1}} +\mathrm{1}}\right)^{\mathrm{3}}…
Question Number 187602 by horsebrand11 last updated on 19/Feb/23 $$\:\:\:\underset{\mathrm{1}/\mathrm{4}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{{x}}\right)−\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{{x}}\right)}{\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{{x}}\right)+\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{{x}}\right)}\:{dx}=? \\ $$ Answered by integralmagic last updated on 19/Feb/23…