Integration Questions

Question Number 124430 by benjo_mathlover last updated on 03/Dec/20

$$\:\int\overset{\:\mathrm{5}} {\:}_{\mathrm{0}} \frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }\:{dx}\:=?\: \\$$

Answered by liberty last updated on 03/Dec/20

$$\mu\:=\underset{\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }\:{dx}\:.\:\left[{use}\:{the}\:{interval}\:\right. \\$$ $$\left.{inverting}\:{substitution}\:{replacing}\:{x}\:{by}\:\mathrm{5}−{x}\:\right] \\$$ $$\mu\:\:=\underset{\mathrm{5}} {\overset{\:\mathrm{0}} {\int}}\:\frac{\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }{\left(\mathrm{5}−{x}\right)^{\mathrm{2}} +{x}^{\mathrm{2}} }\:\left(−{dx}\right)=\underset{\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }\:{dx} \\$$ $${we}\:{get}\:\mathrm{2}\mu\:=\:\underset{\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{{x}^{\mathrm{2}} +\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }{\left(\mathrm{5}−{x}\right)^{\mathrm{2}} +{x}^{\mathrm{2}} }\:{dx}\:=\:\underset{\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:{dx} \\$$ $$\mathrm{2}\mu\:=\:\left({x}\right)_{\mathrm{0}} ^{\mathrm{5}} \:\Rightarrow\mu=\frac{\mathrm{5}}{\mathrm{2}} \\$$