Question Number 186194 by normans last updated on 02/Feb/23
$$ \\ $$$$\:\:\:\:\:\:\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:\:\frac{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:−\:\mathrm{1}}{\mathrm{1}\:+\:\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} }\:\:−\:\:\mathrm{2}}\:\:\boldsymbol{{dx}}\:\:\:\: \\ $$$$ \\ $$
Commented by MJS_new last updated on 02/Feb/23
$$\sqrt{{x}^{\mathrm{2}} }\:\mathrm{why}\:\mathrm{not}\:{x}? \\ $$
Commented by mr W last updated on 02/Feb/23
$${i}\:{don}'{t}\:{think}\:{this}\:{guy}\:{is}\:{serious}\:{or} \\ $$$${really}\:{interested}\:{in}\:{mathemstics}, \\ $$$${when}\:{he}\:{writes}\:{things}\:{like}\:\mathrm{1}+\sqrt{{x}^{\mathrm{2}} }−\mathrm{2}. \\ $$$${maybe}\:{he}\:{just}\:{composes}\:{questions}\: \\ $$$${with}\:{some}\:{mathematical}\:{symbols}, \\ $$$${no}\:{matter}\:{whether}\:{it}\:{makes}\:{sense}\:{or} \\ $$$${not}. \\ $$
Answered by MJS_new last updated on 02/Feb/23
![∫_2 ^4 ((2x^2 −1)/(1+(√x^2 )−2))dx=[x^2 +2x+ln ∣x−∣]_2 ^4 =16+ln 3](Q186205.png)
$$\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} }−\mathrm{2}}{dx}=\left[{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{ln}\:\mid{x}−\mid\right]_{\mathrm{2}} ^{\mathrm{4}} =\mathrm{16}+\mathrm{ln}\:\mathrm{3} \\ $$