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Question Number 118395 by TANMAY PANACEA last updated on 17/Oct/20

$${\int }_a^b\frac{f\left(x\right)}{f\left(x\right)+f(a+b-x)}dx$$

Commented by Dwaipayan Shikari last updated on 17/Oct/20

$$\frac{b-a}{2}$$

Commented by TANMAY PANACEA last updated on 17/Oct/20

$$thankyou...durgapujohochhe$$

Commented by Dwaipayan Shikari last updated on 17/Oct/20

$$From23oct$$

Commented by Dwaipayan Shikari last updated on 17/Oct/20

$$AreyouinKolkatasir?$$

Commented by TANMAY PANACEA last updated on 17/Oct/20

$$naaminagpura..$$

Answered by mathmax by abdo last updated on 17/Oct/20

$$\mathrm{I}={\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})}\mathrm{dx}\mathrm{we}\mathrm{do}\mathrm{the}\mathrm{changement}\mathrm{a}+\mathrm{b}-\mathrm{x}=\mathrm{t}\Rightarrow$$$$\mathrm{I}={\int }_{\mathrm{b}}^{\mathrm{a}}\frac{\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{t})}{\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{t})+\mathrm{f}\left(\mathrm{t}\right)}(-\mathrm{dt})={\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})}{\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx}\Rightarrow$$$$2\mathrm{I}={\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})}{\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{dx}=\mathrm{b}-\mathrm{a}\Rightarrow \mathrm{I}=\frac{\mathrm{b}-\mathrm{a}}{2}$$

Commented by TANMAY PANACEA last updated on 17/Oct/20

$$thankyousir$$

Commented by Bird last updated on 17/Oct/20

$$youstillworkatrailwayssirtanmay..$$

Commented by TANMAY PANACEA last updated on 17/Oct/20

$$yessirtillbecomeseniorcitizeninyear2030$$

Commented by mathmax by abdo last updated on 17/Oct/20

$$\mathrm{good}\mathrm{luck}\mathrm{sir}\mathrm{tanmay}$$

Answered by TANMAY PANACEA last updated on 17/Oct/20

$$formula$$$${\int }_q^{p}f\left(x\right)dx={\int }_q^{p}f(p+q-x)dx$$$$I={\int }_a^b\frac{f\left(x\right)}{f\left(x\right)+f(a+b-x)}dx$$$$={\int }_a^b\frac{f(a+b-x)}{f(a+b-x)+f\{(a+b-(a+b-x)\}}dx$$$$={\int }_a^b\frac{f(a+b-x)}{f(a+b-x)+f\left(x\right)}dx$$$$2I={\int }_a^{b}dx$$$$I=\frac{b-a}{2}$$$$$$

Commented by bramlexs22 last updated on 17/Oct/20

$$thisisformulaKingIntegralsir?$$

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