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Category: Integration

f-x-2-2-0-x-2-f-y-1-tan-y-dy-x-R-f-pi-

Question Number 161100 by bobhans last updated on 13/Dec/21 $$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}−\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(−\pi\right)=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

show-that-sin-x-sin-x-dx-xcos-x-sin-log-sin-x-

Question Number 95562 by peter frank last updated on 26/May/20 $$\mathrm{show}\:\mathrm{that} \\ $$$$\int\frac{\mathrm{sin}\:\left(\mathrm{x}−\theta\right)}{\mathrm{sin}\:\mathrm{x}}\mathrm{dx}=\mathrm{xcos}\:\mathrm{x}−\mathrm{sin}\:\theta\mathrm{log}\:\mathrm{sin}\:\mathrm{x} \\ $$ Answered by Ar Brandon last updated on 26/May/20 $$\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\theta}\right)=\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\theta}\right)−\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\theta}\right) \\…

ax-2-bx-c-x-p-x-q-x-r-dx-

Question Number 95563 by peter frank last updated on 26/May/20 $$\int\frac{\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}}{\left(\mathrm{x}−\mathrm{p}\right)\left(\mathrm{x}−\mathrm{q}\right)\left(\mathrm{x}−\mathrm{r}\right)}\mathrm{dx} \\ $$ Answered by MJS last updated on 26/May/20 $$\int\frac{{ax}^{\mathrm{2}} +{bx}+{c}}{\left({x}−{p}\right)\left({x}−{q}\right)\left({x}−{r}\right)}{dx}= \\ $$$$=\frac{{ap}^{\mathrm{2}}…

prove-that-I-0-pi-2-ln-1-sin-2-d-2G-pi-ln-2-G-catalan-constant-

Question Number 161089 by mnjuly1970 last updated on 12/Dec/21 $$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:−\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$ Answered by Ar…