Question Number 83675 by niroj last updated on 05/Mar/20
$$ \\ $$$$\: \\ $$$$\:\:\mathrm{evaluate}: \\ $$$$\:\mathrm{2}\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\frac{\sqrt{\mathrm{x}+\mathrm{1}}}{\mathrm{x}^{\mathrm{2}} +\mathrm{4}}\mathrm{dx} \\ $$$$\:\:\:\:\: \\ $$$$\:\: \\ $$
Commented by jagoll last updated on 05/Mar/20
$$\sqrt{{x}+\mathrm{1}}\:=\:{u}\:\Rightarrow\:{x}^{\mathrm{2}} =\:\left({u}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\mathrm{2}{x}\:{dx}\:=\:\mathrm{4}{u}\:\left({u}^{\mathrm{2}} −\mathrm{1}\right){du} \\ $$$${dx}\:=\:\frac{\mathrm{2}{u}\left({u}^{\mathrm{2}} −\mathrm{1}\right){du}}{{u}^{\mathrm{2}} −\mathrm{1}}\:=\:\mathrm{2}{u}\:{du} \\ $$$${x}^{\mathrm{2}} \:+\mathrm{4}\:=\:\left({u}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} +\mathrm{4} \\ $$$$\mathrm{2}\:\overset{\sqrt{\mathrm{3}}} {\int}_{\mathrm{1}} \:\frac{{u}\:\left(\mathrm{2}{u}\:{du}\right)}{\mathrm{4}+\left({u}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$$${let}\:{u}\:=\:\mathrm{sec}\:{t}\:\Rightarrow\:\mathrm{du}\:=\:\mathrm{sec}\:{t}\:\mathrm{tan}\:{t}\:{dt} \\ $$$$ \\ $$
Commented by niroj last updated on 05/Mar/20
$${thanks}\:{mr}.{jagoll}\:{try}\:{to}\:{solve}\:{if}\:{it}\:{is}\:{better}\:{tobe}\:{complete}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Answered by TANMAY PANACEA last updated on 05/Mar/20
$${I}=\int\frac{{x}+\mathrm{1}}{\:\sqrt{{x}+\mathrm{1}}\:×\left({x}^{\mathrm{2}} +\mathrm{4}\right)}{dx} \\ $$$${t}^{\mathrm{2}} ={x}+\mathrm{1} \\ $$$$\int\frac{{t}^{\mathrm{2}} ×\mathrm{2}{tdt}}{{t}\left({t}^{\mathrm{4}} −\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}+\mathrm{4}\right)} \\ $$$$\mathrm{2}\int\frac{{t}^{\mathrm{2}} }{{t}^{\mathrm{4}} −\mathrm{2}{t}^{\mathrm{2}} +\mathrm{5}}{dt} \\ $$$$\mathrm{2}\int\frac{{dt}}{{t}^{\mathrm{2}} +\frac{\mathrm{5}}{{t}^{\mathrm{2}} }−\mathrm{2}} \\ $$$${t}^{\mathrm{2}} +\frac{\mathrm{5}}{{t}^{\mathrm{2}} }=\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)^{\mathrm{2}} −\mathrm{2}\sqrt{\mathrm{5}}\: \\ $$$$\frac{{d}}{{dt}}\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)=\mathrm{1}−\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} } \\ $$$${t}^{\mathrm{2}} +\frac{\mathrm{5}}{{t}^{\mathrm{2}} }=\left({t}−\frac{\sqrt{\mathrm{5}}}{{t}}\right)^{\mathrm{2}} +\mathrm{2}\sqrt{\mathrm{5}}\: \\ $$$$\frac{{d}}{{dt}}\left({t}−\frac{\sqrt{\mathrm{5}}}{{t}}\right)=\mathrm{1}+\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} } \\ $$$${look}\:\:\left(\mathrm{1}+\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} }\right)+\left(\mathrm{1}−\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} }\right)=\mathrm{2} \\ $$$${now}\int\frac{\mathrm{1}−\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} }+\mathrm{1}+\frac{\sqrt{\mathrm{5}}}{{t}^{\mathrm{2}} }}{{t}^{\mathrm{2}} +\frac{\mathrm{5}}{{t}^{\mathrm{2}} }−\mathrm{2}}{dt} \\ $$$$\int\frac{{d}\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)}{\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)^{\mathrm{2}} −\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}}+\int\frac{{d}\left({t}−\frac{\sqrt{\mathrm{5}}}{{t}}\right)}{\left({t}−\frac{\sqrt{\mathrm{5}}}{{t}}\right)^{\mathrm{2}} +\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}} \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{pls}}\:\boldsymbol{{use}}\:\boldsymbol{{formula}}\:\boldsymbol{{of}}\: \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\:\:{and}\:\int\frac{{dx}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} } \\ $$$$\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{2}=\left(\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{2}}\:\right)^{\mathrm{2}} \:\:\leftarrow{as}\:{if}\:{a}^{\mathrm{2}} \\ $$$$\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}=\left(\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}}\:\right)^{\mathrm{2}} \\ $$$${so} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\left(\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{2}}\:\right)}{ln}\left(\frac{\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)−\left(\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{2}}\:\right)}{\left({t}+\frac{\sqrt{\mathrm{5}}}{{t}}\right)+\left(\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:+\mathrm{2}}\:\right)}\right)+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}}}{tan}^{−\mathrm{1}} \left(\frac{{t}−\frac{\sqrt{\mathrm{5}}}{{t}}}{\:\sqrt{\mathrm{2}\sqrt{\mathrm{5}}\:−\mathrm{2}}}\right) \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{pls}}\:\boldsymbol{{replace}}\:\boldsymbol{{t}}\:\boldsymbol{{by}}\:\sqrt{\boldsymbol{{x}}+\mathrm{1}}\:\boldsymbol{{and}}\:\boldsymbol{{put}}\:\boldsymbol{{upper}}\: \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{lower}}\:\boldsymbol{{limit}} \\ $$$$ \\ $$
Commented by niroj last updated on 05/Mar/20
$${you}\:{are}\:{absute}\:{amazing}.{nice}\:{solution}.. \\ $$
Commented by TANMAY PANACEA last updated on 05/Mar/20
$${thank}\:{you}\:{sir} \\ $$