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Category: Integration

Prove-that-6-11-dx-x-2-x-3-2ln-3-2-2-2-3-

Question Number 17472 by Arnab Maiti last updated on 06/Jul/17 $$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{6}} ^{\mathrm{11}} \frac{\mathrm{dx}}{\:\sqrt{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}}=\mathrm{2ln}\frac{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$ Answered by sma3l2996 last updated on 06/Jul/17 $${I}=\int_{\mathrm{6}} ^{\mathrm{11}} \frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}}…

cos-4x-cos-2x-sin-4x-cos-2x-dx-

Question Number 82954 by john santu last updated on 26/Feb/20 $$\int\:\frac{\mathrm{cos}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{sin}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\: \\ $$ Commented by john santu last updated on 26/Feb/20 $$\mathrm{let}\:\mathrm{u}=\:\mathrm{cos}\:\mathrm{2x}\:\Rightarrow\:\mathrm{dx}\:=\:−\frac{\mathrm{du}}{\mathrm{2}\sqrt{\mathrm{1}−\mathrm{u}^{\mathrm{2}} }} \\ $$$$\Rightarrow\int\:\frac{\:\mathrm{2u}^{\mathrm{2}}…

Question-148467

Question Number 148467 by bramlexs22 last updated on 28/Jul/21 Answered by puissant last updated on 28/Jul/21 $$\mathrm{x}=\mathrm{tan}\left(\mathrm{t}\right)\Rightarrow\mathrm{dx}=\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{0}\leqslant\mathrm{x}\leqslant\infty\:\Rightarrow\:\mathrm{0}\leqslant\mathrm{t}\leqslant\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{arctan}\left(\mathrm{tan}\left(\mathrm{t}\right)\right)}{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\mathrm{t}\right)}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}}…