Question Number 54826 by rahul 19 last updated on 13/Feb/19
$$\left.\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} −\left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} \right){dx}\:=\:? \\ $$$$\left.\mathrm{2}\right)\:{If}\:{f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{4}\:{then}\:{the}\:{value}\:{of} \\ $$$$\:\int_{−\mathrm{1}} ^{\:\mathrm{1}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{0}} ^{\:\mathrm{4}} {f}^{\:−\mathrm{1}} \left({x}\right){dx}\:=\:? \\ $$$$\left.\mathrm{3}\right)\:\int_{−\pi} ^{\:\pi} \left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right)\left(\mathrm{1}+{sinx}+…+{sin}\mathrm{13}{x}\right){dx}=? \\ $$$$\left.\mathrm{4}\right)\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\left(\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:+\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:\right){dx}\:=? \\ $$$$\left({This}\:{time}\:{func}.\:{are}\:{not}\:{inverse}\:{of}\:{each}\right. \\ $$$$\left.{other},{right}\:?\right) \\ $$
Commented by maxmathsup by imad last updated on 12/Feb/19
$$\left.\mathrm{2}\right)\:{we}\:{have}\:\int_{−\mathrm{1}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\int_{−\mathrm{1}} ^{\mathrm{1}} \left({x}^{\mathrm{3}} \:+\mathrm{3}{x}+\mathrm{4}\right){dx}\:=\left[\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:+\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} \:+\mathrm{4}{x}\right]_{−\mathrm{1}} ^{\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\:+\frac{\mathrm{3}}{\mathrm{2}}\:+\mathrm{4}\:−\frac{\mathrm{1}}{\mathrm{4}}\:−\frac{\mathrm{3}}{\mathrm{2}}\:+\mathrm{4}\:=\mathrm{8}\:\:\:{let}\:{find}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$${f}\left({x}\right)={y}\:\Leftrightarrow{x}^{\mathrm{3}} \:+\mathrm{3}{x}\:+\mathrm{4}\:={y}\:\Leftrightarrow{x}^{\mathrm{3}} \:+\mathrm{3}{x}+\mathrm{4}−{y}\:=\mathrm{0}\:\:\:{changement}\:{x}\:={u}+{v}\:{give} \\ $$$${u}^{\mathrm{3}} \:+\mathrm{3}{uv}\left({u}+{v}\right)\:+{v}^{\mathrm{3}} \:\:+\mathrm{3}\left({u}+{v}\right)\:+\mathrm{4}−{y}\:=\mathrm{0}\:\Rightarrow \\ $$$${u}^{\mathrm{3}} \:+{v}^{\mathrm{3}} \:+\mathrm{4}−{y}\:+\:\left({u}+{v}\right)\left(\mathrm{3}{uv}+\mathrm{3}\right)\:=\mathrm{0}\:\Rightarrow{u}^{\mathrm{3}} \:+{v}^{\mathrm{3}} \:={y}−\mathrm{4}\:{and}\:{uv}\:=−\mathrm{1}\:\Rightarrow \\ $$$${so}\:{u}^{\mathrm{3}} \:{and}\:{v}^{\mathrm{3}} \:{are}\:{solution}\:{of}\:{the}\:{equation}\:{X}^{\mathrm{2}} −\left({y}−\mathrm{4}\right){X}\:−\mathrm{1}\:=\mathrm{0}\:\Rightarrow \\ $$$$\Delta\:=\left({y}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}>\mathrm{0}\:\Rightarrow{X}_{\mathrm{1}} =\frac{{y}−\mathrm{4}\:+\sqrt{\left({y}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}}}{\mathrm{2}}\:\:{and} \\ $$$${X}_{\mathrm{2}} =\frac{{y}−\mathrm{4}\:−\sqrt{\left.\left({y}−\mathrm{4}\right)^{\mathrm{2}} \:\right)\mathrm{4}}}{\mathrm{2}}\:\:\Rightarrow{x}\:=^{\mathrm{3}} \sqrt{{X}_{\mathrm{1}} }\:+\sqrt{{X}_{\mathrm{2}} } \\ $$$$\Rightarrow{f}^{−\mathrm{1}} \left({x}\right)=\left(\frac{{x}−\mathrm{4}+\sqrt{\left({x}−\mathrm{4}\right)^{\mathrm{2}} +\mathrm{4}}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\left(\frac{{x}−\mathrm{4}\:−\sqrt{\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}} \:{f}^{−\mathrm{1}} \left({x}\right){dx}\:=\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)}\int_{\mathrm{0}} ^{\mathrm{4}} \:\left({x}−\mathrm{4}+\sqrt{\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}\:+\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)}\int_{\mathrm{0}} ^{\mathrm{4}} \left({x}−\mathrm{4}−\sqrt{\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx} \\ $$$${let}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{4}} \left({x}−\mathrm{4}\:+\sqrt{\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:+\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}\:\:\:\:{chang}\:.{x}−\mathrm{4}\:=\mathrm{2}{sh}\left({t}\right)\:{give} \\ $$$${I}\:=\:\int_{−{argsh}\left(\mathrm{2}\right)} ^{\mathrm{0}} \:\left(\mathrm{2}{sh}\left({t}\right)\:+\mathrm{2}\:{ch}\left({t}\right)\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:\mathrm{2}{ch}\left({t}\right){dt} \\ $$$$=\mathrm{2}\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)\:\:\int_{−{ln}\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)} ^{\mathrm{0}} \:\left({sh}\left({t}\right)+{ch}\left({t}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:{ch}\left({t}\right){dt}\:\:….{be}\:{continued}…\right. \\ $$$${let}\:{A}\:=\int_{\mathrm{0}} ^{\mathrm{4}} \:{f}^{−\mathrm{1}} \left({x}\right){dx}\:\:{changement}\:{f}^{−\mathrm{1}} \left({x}\right)=\:{t}\:{give}\:{x}={f}\left({t}\right)\:\Rightarrow \\ $$$${A}\:=\int_{{f}^{−\mathrm{1}} \left(\mathrm{0}\right)} ^{{f}^{−\mathrm{1}} \left(\mathrm{4}\right)} \:\:{t}\:{f}^{'} \left({t}\right){dt}\:=\:\left[{t}\:{f}\left({t}\right)\right]_{{f}^{−\mathrm{1}} \left(\mathrm{0}\right)} ^{{f}^{−\mathrm{1}} \left(\mathrm{4}\right)} \:−\int_{{f}^{−\mathrm{1}} \left(\mathrm{0}\right)} ^{{f}^{−\mathrm{1}} \left(\mathrm{4}\right)} \:{f}\left({t}\right){dt} \\ $$$${but}\:{f}^{−\mathrm{1}} \left(\mathrm{4}\right)\:=\mathrm{1}\:−\mathrm{1}\:=\mathrm{0}\:\:{and}\:{f}^{−\mathrm{1}} \left(\mathrm{0}\right)\:=\left(\frac{−\mathrm{4}\:+\sqrt{\mathrm{20}}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\left(\frac{−\mathrm{4}−\sqrt{\mathrm{20}}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$=\left(−\mathrm{2}+\sqrt{\mathrm{5}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:−\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)^{\frac{\mathrm{1}}{\mathrm{3}\:}} \:\:=\alpha_{\mathrm{0}} \:\Rightarrow \\ $$$${A}\:=\left[{tf}\left({t}\right)\right]_{\alpha_{\mathrm{0}} } ^{\mathrm{0}} \:−\int_{\alpha_{\mathrm{0}} } ^{\mathrm{0}} \:{f}\left({t}\right){dt}\:=−\alpha_{\mathrm{0}} {f}\left(\alpha_{\mathrm{0}} \right)\:+\int_{\mathrm{0}} ^{\alpha_{\mathrm{0}} } \left(\:{t}^{\mathrm{3}} \:+\mathrm{3}{t}+\mathrm{4}\right){dt} \\ $$$$=−\alpha_{\mathrm{0}} {f}\left(\alpha_{\mathrm{0}} \right)\:+\frac{\alpha_{\mathrm{0}} ^{\mathrm{3}} }{\mathrm{3}}\:+\frac{\mathrm{3}}{\mathrm{2}}\:\alpha_{\mathrm{0}} ^{\mathrm{2}} \:+\mathrm{4}\alpha_{\mathrm{0}} \\ $$$${so}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{4}} \:{f}^{−\mathrm{1}} \left({x}\right)\:{dx}\:\:{is}\:{known}\:. \\ $$
Commented by maxmathsup by imad last updated on 12/Feb/19
$${in}\:{general}\:{let}\:{find}\:\int_{{a}} ^{{b}} \:{f}^{−\mathrm{1}} \left({x}\right){dx}\:\:{chang}.{f}^{−\mathrm{1}} \left({x}\right)\:={t}\:{give}\:{x}\:={f}\left({t}\right)\:\Rightarrow \\ $$$$\int_{{a}} ^{{b}} \:{f}^{−\mathrm{1}} \left({x}\right){dx}\:=\int_{{f}^{−\mathrm{1}} \left({a}\right)} ^{{f}^{−\mathrm{1}} \left({b}\right)} \:{t}\:{f}^{'} \left({t}\right){dt}\:=\left[{t}\:{f}\left({t}\right)\right]_{{f}^{−\mathrm{1}} \left({a}\right)} ^{{f}^{−\mathrm{1}} \left({b}\right)} \:−\int_{{f}^{−\mathrm{1}} \left({a}\right)} ^{{f}^{−\mathrm{1}} \left({b}\right)} \:{f}\left({t}\right){dt} \\ $$$$={bf}^{−\mathrm{1}} \left({b}\right)−{af}^{−\mathrm{1}} \left({a}\right)\:−\int_{{f}^{−\mathrm{1}} \left({a}\right)} ^{{f}^{−\mathrm{1}} \left({b}\right)} {f}\left({t}\right){dt}\:. \\ $$
Commented by kaivan.ahmadi last updated on 12/Feb/19
$$\mathrm{y}=\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\Rightarrow\mathrm{f}\left(\mathrm{y}\right)=\mathrm{x}\Rightarrow\mathrm{y}^{\mathrm{3}} +\mathrm{3y}+\mathrm{4}=\mathrm{x},\mathrm{dx}=\left(\mathrm{3y}^{\mathrm{2}} +\mathrm{3}\right)\mathrm{dy} \\ $$$$\mathrm{if}\:\mathrm{x}=\mathrm{0}\Rightarrow\mathrm{y}=−\mathrm{1} \\ $$$$\mathrm{if}\:\mathrm{x}=\mathrm{4}\Rightarrow\mathrm{y}=\mathrm{0}\:\mathrm{or}\:−\mathrm{3}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}} \mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\mathrm{dx}=\int_{−\mathrm{1}} ^{\mathrm{0}} \mathrm{y}\left(\mathrm{3y}^{\mathrm{2}} +\mathrm{3}\right)\mathrm{dy}=\left[\frac{\mathrm{3y}^{\mathrm{4}} }{\mathrm{4}}+\frac{\mathrm{3y}^{\mathrm{2}} }{\mathrm{2}}\right]_{−\mathrm{1}} ^{\mathrm{0}} =−\left(\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{3}}{\mathrm{2}}\right)=−\frac{\mathrm{9}}{\mathrm{4}} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{true}? \\ $$$$ \\ $$
Commented by maxmathsup by imad last updated on 12/Feb/19
$$\left.\mathrm{3}\right)\:{let}\:{simlify}\:{A}\left({x}\right)=\left(\mathrm{1}+{cosx}\:+{cos}\left(\mathrm{2}{x}\right)+…+{cos}\left(\mathrm{3}{x}\right)\right)\left(\mathrm{1}+{sinx}\:+{sin}\left(\mathrm{2}{x}\right)+…+{sin}\left(\mathrm{13}{x}\right)\right) \\ $$$${let}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {cos}\left({kx}\right)\:{and}\:{W}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{sin}\left({kx}\right)\:\Rightarrow\:{S}_{{n}} \:+{iW}_{{n}} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}} \:\left({e}^{{ix}} \right)^{{k}} \:=\frac{\mathrm{1}−\left({e}^{{ix}} \right)^{{n}+\mathrm{1}} }{\mathrm{1}−{e}^{{ix}} }\:=\:\frac{\mathrm{1}−{e}^{{i}\left({n}+\mathrm{1}\right){x}} }{\mathrm{1}−{cosx}\:−{isinx}} \\ $$$$=\frac{\mathrm{1}−{cos}\left({n}+\mathrm{1}\right){x}\:−{isin}\left({n}+\mathrm{1}\right){x}}{\mathrm{1}−{cosx}\:−{isinx}}\:=\frac{\mathrm{2}{sin}^{\mathrm{2}} \frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}−\mathrm{2}{i}\:{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right){cos}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right)}{\mathrm{2}\:{sin}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{2}{i}\:{sin}\left(\frac{{x}}{\mathrm{2}}\right){cos}\left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\frac{−{i}\:{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right)\:{e}^{{i}\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}} }{−{isin}\left(\frac{{x}}{\mathrm{2}}\right)\:{e}^{\frac{{ix}}{\mathrm{2}}} }\:=\frac{{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right){e}^{{i}\frac{{nx}}{\mathrm{2}}} }{{sin}\left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\frac{{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\left\{\:{cos}\left(\frac{{nx}}{\mathrm{2}}\right)\:+{isin}\left(\frac{{nx}}{\mathrm{2}}\right)\right\}\:\Rightarrow \\ $$$$\:{S}_{{n}} \left({x}\right)\:=\:\frac{{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right)\:{cos}\left(\frac{{nx}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:{and}\:{W}_{{n}} \left({x}\right)=\frac{{sin}\left(\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}\right){sin}\left(\frac{{nx}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:\Rightarrow \\ $$$$\mathrm{1}+{cosx}\:+{cos}\left(\mathrm{2}{x}\right)+…{cos}\left(\mathrm{13}{x}\right)=\frac{{sin}\left(\mathrm{7}{x}\right){cos}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:{and}\: \\ $$$$\mathrm{1}+{sinx}\:+{sin}\left(\mathrm{2}{x}\right)+…+{sin}\left(\mathrm{13}{x}\right)\:=\mathrm{1}+\frac{{sin}\left(\mathrm{7}{x}\right){sin}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:\Rightarrow \\ $$$$\left(…\right)×\left(…\right)\:=\frac{{sin}\left(\mathrm{7}{x}\right)\:{cos}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:+\frac{{sin}^{\mathrm{2}} \left(\mathrm{7}{x}\right){sin}\left(\mathrm{13}{x}\right)}{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}\:\:\Rightarrow \\ $$$$\int_{−\pi} ^{\pi} \:{A}\left({x}\right){dx}\:=\:\int_{−\pi} ^{\pi} \:\frac{{sin}\left(\mathrm{7}{x}\right){cos}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}{dx}\:\:+\int_{−\pi} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} \left(\mathrm{7}{x}\right){sin}\left(\mathrm{13}{x}\right)}{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{dx} \\ $$$$\int_{−\pi} ^{\pi} \:\:\frac{{sin}\left(\mathrm{7}{x}\right){cos}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}\:{dx}\:=\mathrm{2}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{sin}\left(\mathrm{7}{x}\right){cos}\left(\frac{\mathrm{13}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}{dx}\:\left({even}\:{function}\right) \\ $$$$\int_{−\pi} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} \left(\mathrm{7}{x}\right){sin}\left(\mathrm{13}{x}\right)}{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}{dx}\:=\mathrm{0}\:\left(\:{odd}\:{function}\right)\:…{be}\:{continued}…. \\ $$$$ \\ $$
Commented by Meritguide1234 last updated on 13/Feb/19
Answered by tanmay.chaudhury50@gmail.com last updated on 12/Feb/19
$$\left.\mathrm{1}\right)\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} {dx}−\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dx} \\ $$$${I}_{\mathrm{1}} −{I}_{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{{a}} \right)^{\frac{\mathrm{1}}{{b}}} {dx} \\ $$$${x}^{{a}} ={sin}^{\mathrm{2}} \alpha\:\:{x}=\left({sin}\alpha\right)^{\frac{\mathrm{2}}{{a}}} \\ $$$${dx}=\frac{\mathrm{2}}{{a}}×\left({sin}\alpha\right)^{\frac{\mathrm{2}}{{a}}−\mathrm{1}} \left({cos}\alpha\right){d}\alpha \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−{sin}^{\mathrm{2}} \alpha\right)^{\frac{\mathrm{1}}{{b}}} ×\frac{\mathrm{2}}{{a}}×\left({sin}\alpha\right)^{\frac{\mathrm{2}}{{a}}−\mathrm{1}} \left({cos}\alpha\right){d}\alpha \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({cos}\alpha\right)^{\frac{\mathrm{2}}{{b}}} ×\frac{\mathrm{2}}{{a}}×\left({sin}\alpha\right)^{\frac{\mathrm{2}}{{a}}−\mathrm{1}} \left({cos}\alpha\right){d}\alpha \\ $$$$\frac{\mathrm{2}}{{a}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({sin}\alpha\right)^{\frac{\mathrm{2}}{{a}}−\mathrm{1}} ×\left({cos}\alpha\right)^{\frac{\mathrm{2}}{{b}}+\mathrm{1}} {d}\alpha \\ $$$$\frac{\mathrm{1}}{{a}}×\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sin}\alpha\right)^{\mathrm{2}×\frac{\mathrm{1}}{{a}}−\mathrm{1}} \left({cos}\alpha\right)^{\mathrm{2}\left(\frac{\mathrm{1}}{{b}}+\mathrm{1}\right)−\mathrm{1}} {d}\alpha \\ $$$${formula} \\ $$$$\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sin}\alpha\right)^{\mathrm{2}{p}−\mathrm{1}} \left({cos}\alpha\right)^{\mathrm{2}{q}−\mathrm{1}} {d}\alpha \\ $$$$\frac{\left.\lceil\left({p}\right)\lceil{q}\right)}{\lceil\left({p}+{q}\right)} \\ $$$${so}\:\frac{\mathrm{1}}{{a}}×\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sin}\alpha\right)^{\mathrm{2}×\frac{\mathrm{1}}{{a}}−\mathrm{1}} \left({cos}\alpha\right)^{\mathrm{2}\left(\frac{\mathrm{1}}{{b}}+\mathrm{1}\right)−\mathrm{1}} {d}\alpha \\ $$$$=\frac{\mathrm{1}}{{a}}×\frac{\lceil\left(\frac{\mathrm{1}}{{a}}\right)\lceil\left(\frac{\mathrm{1}}{{b}}+\mathrm{1}\right)}{\lceil\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\mathrm{1}\right)} \\ $$$${so}\:\:{I}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{7}}×\frac{\lceil\left(\frac{\mathrm{1}}{\mathrm{7}}\right)\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{1}\right)}{\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{7}}+\mathrm{1}\right)}\:\left[{a}=\mathrm{7}\:\:\:{b}=\mathrm{4}\right] \\ $$$${I}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}}×\frac{\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}\right)\lceil\left(\frac{\mathrm{1}}{\mathrm{7}}+\mathrm{1}\right)}{\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{7}}+\mathrm{1}\right)}\:\left[{a}=\mathrm{4}\:\:\:{b}=\mathrm{7}\right] \\ $$$${required}\:{ans}={I}_{\mathrm{1}} −{I}_{\mathrm{2}} \\ $$$${formula}\:\lceil\left({n}+\mathrm{1}\right)={n}! \\ $$$${pls}\:{check}\:{upto}\:{this}\:{step}… \\ $$$$ \\ $$$${I}_{\mathrm{1}} −{I}_{\mathrm{2}} =\frac{\mathrm{1}}{\lceil\left(\frac{\mathrm{11}}{\mathrm{28}}+\mathrm{1}\right)}\left[\frac{\mathrm{1}}{\mathrm{7}}×\lceil\left(\frac{\mathrm{1}}{\mathrm{7}}\right)×\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{1}\right)−\frac{\mathrm{1}}{\mathrm{4}}×\lceil\left(\frac{\mathrm{1}}{\mathrm{4}}\right)×\lceil\left(\frac{\mathrm{1}}{\mathrm{7}}+\mathrm{1}\right)\right] \\ $$$${let}\:{a}=\frac{\mathrm{1}}{\mathrm{7}}\:\:\:{b}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.{ans}=\frac{\mathrm{1}}{\lceil\left({a}+{b}+\mathrm{1}\right)}×\left[{a}×\lceil{a}\right)\lceil\left({b}+\mathrm{1}\right)−{b}\lceil\left({b}\right)\lceil\left({a}+\mathrm{1}\right)\right] \\ $$$$=\frac{\mathrm{1}}{\left({a}+{b}\right)!}×\left[{a}\left({a}−\mathrm{1}\right)!{b}!−{b}\left({b}−\mathrm{1}\right)!{a}!\right] \\ $$$$=\frac{\mathrm{1}}{\left({a}+{b}\right)!}\left[{a}!{b}!−{b}!{a}!\right]=\mathrm{0} \\ $$
Commented by mr W last updated on 12/Feb/19
$${f}\left({x}\right)=\left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} ={y} \\ $$$$\Rightarrow{x}=\left(\mathrm{1}−{y}^{\mathrm{4}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$\Rightarrow{f}^{−\mathrm{1}} \left({x}\right)=\left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} ={g}\left({x}\right) \\ $$$$\Rightarrow{g}\left({x}\right)\:{is}\:{inverse}\:{function}\:{of}\:{f}\left({x}\right),\:{i}.{e}. \\ $$$${g}\left({f}\left({x}\right)\right)={x} \\ $$$${I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} {g}\left({y}\right){dy} \\ $$$${let}\:{y}={f}\left({x}\right) \\ $$$$\Rightarrow{g}\left({y}\right)={g}\left({f}\left({x}\right)\right)={x} \\ $$$${dy}={f}'\left({x}\right){dx} \\ $$$$\Rightarrow{I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} {g}\left({y}\right){dy}=\int_{{g}\left(\mathrm{0}\right)} ^{{g}\left(\mathrm{1}\right)} {xf}'\left({x}\right){dx}=\int_{\mathrm{1}} ^{\mathrm{0}} {xf}'\left({x}\right){dx}=−\int_{\mathrm{0}} ^{\mathrm{1}} {xf}'\left({x}\right){dx} \\ $$$${I}_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$${I}={I}_{\mathrm{1}} −{I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}+\int_{\mathrm{0}} ^{\mathrm{1}} {xf}'\left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left[{f}\left({x}\right)+{xf}'\left({x}\right)\right]{dx} \\ $$$$=\left[{xf}\left({x}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\mathrm{0} \\ $$
Commented by rahul 19 last updated on 12/Feb/19
$${Sir},{I}\:'{m}\:{new}\:{to}\:{this}\:{function}! \\ $$
Commented by rahul 19 last updated on 12/Feb/19
Commented by rahul 19 last updated on 12/Feb/19
$${this}\:{is}\:{sol}^{{n}} {given}.\:{Pl}\:{explain}\:{how} \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} {g}\left({y}\right){dy}\:=\:\int_{\mathrm{1}} ^{\mathrm{0}} {xf}\:'\left({x}\right){dx}\:? \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 12/Feb/19
$${excellent}\:{sir}… \\ $$
Commented by rahul 19 last updated on 13/Feb/19
$${Thank}\:{you}\:{sir}! \\ $$
Answered by mr W last updated on 12/Feb/19
$${Q}\left(\mathrm{2}\right) \\ $$$${f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{4} \\ $$$${I}_{\mathrm{1}} =\int_{−\mathrm{1}} ^{\mathrm{1}} {f}\left({x}\right){dx}=\int_{−\mathrm{1}} ^{\mathrm{0}} {f}\left({x}\right){dx}+\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$${I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{4}} {f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$${let}\:{x}={f}\left({t}\right)\Rightarrow{t}={f}^{−\mathrm{1}} \left({x}\right) \\ $$$${dx}={f}'\left({t}\right){dt} \\ $$$${I}_{\mathrm{2}} =\int_{{f}^{−\mathrm{1}} \left(\mathrm{0}\right)} ^{{f}^{−\mathrm{1}} \left(\mathrm{4}\right)} {tf}'\left({t}\right){dt}=\int_{−\mathrm{1}} ^{\mathrm{0}} {tf}'\left({t}\right){dt}=\int_{−\mathrm{1}} ^{\mathrm{0}} {xf}'\left({x}\right){dx} \\ $$$${I}={I}_{\mathrm{1}} +{I}_{\mathrm{2}} =\int_{−\mathrm{1}} ^{\mathrm{0}} {f}\left({x}\right){dx}+\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}+\int_{−\mathrm{1}} ^{\mathrm{0}} {xf}'\left({x}\right){dx} \\ $$$$=\int_{−\mathrm{1}} ^{\mathrm{0}} \left[{f}\left({x}\right)+{xf}'\left({x}\right)\right]{dx}+\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$=\left[{xf}\left({x}\right)\right]_{−\mathrm{1}} ^{\mathrm{0}} +\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$=\mathrm{0}+\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{4}\right){dx} \\ $$$$=\left[\frac{{x}^{\mathrm{4}} }{\mathrm{4}}+\frac{\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}}+\mathrm{4}{x}\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{4} \\ $$$$=\frac{\mathrm{23}}{\mathrm{4}} \\ $$$$\:\Rightarrow\int_{−\mathrm{1}} ^{\:\mathrm{1}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{0}} ^{\:\mathrm{4}} {f}^{\:−\mathrm{1}} \left({x}\right){dx}\:=\frac{\mathrm{23}}{\mathrm{4}} \\ $$
Commented by mr W last updated on 12/Feb/19
$${certainly}\:{one}\:{can}\:{also}\:{do}\:{like}\:{this}: \\ $$$${I}_{\mathrm{1}} =\int_{−\mathrm{1}} ^{\mathrm{1}} {f}\left({x}\right){dx}=\int_{−\mathrm{1}} ^{\mathrm{1}} \left({x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{4}\right){dx} \\ $$$$=\left[\frac{{x}^{\mathrm{4}} }{\mathrm{4}}+\frac{\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}}+\mathrm{4}{x}\right]_{−\mathrm{1}} ^{\mathrm{1}} \\ $$$$=\mathrm{8} \\ $$$${I}_{\mathrm{2}} =\int_{{f}^{−\mathrm{1}} \left(\mathrm{0}\right)} ^{{f}^{−\mathrm{1}} \left(\mathrm{4}\right)} {tf}'\left({t}\right){dt}=\int_{−\mathrm{1}} ^{\mathrm{0}} {tf}'\left({t}\right){dt}=\int_{−\mathrm{1}} ^{\mathrm{0}} {t}\left(\mathrm{3}{t}^{\mathrm{2}} +\mathrm{3}\right){dt} \\ $$$$=\mathrm{3}\int_{−\mathrm{1}} ^{\mathrm{0}} \left({t}^{\mathrm{3}} +{t}\right){dt} \\ $$$$=\mathrm{3}\left[\frac{{t}^{\mathrm{4}} }{\mathrm{4}}+\frac{{t}^{\mathrm{2}} }{\mathrm{2}}\right]_{−\mathrm{1}} ^{\mathrm{0}} \\ $$$$=−\mathrm{3}\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=−\frac{\mathrm{9}}{\mathrm{4}} \\ $$$$\Rightarrow{I}=\mathrm{8}−\frac{\mathrm{9}}{\mathrm{4}}=\frac{\mathrm{23}}{\mathrm{4}} \\ $$
Commented by rahul 19 last updated on 13/Feb/19
$${yes}\:,\:\mathrm{2}{nd}\:{method}\:{works}\:{only}\:{when}\:{fun}^{{c}} \\ $$$${is}\:{easy}\:{to}\:{integrate}! \\ $$
Answered by mr W last updated on 12/Feb/19
$${Q}\left(\mathrm{3}\right) \\ $$$${I}=\int_{−\pi} ^{\:\pi} \left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right)\left(\mathrm{1}+{sinx}+…+{sin}\mathrm{13}{x}\right){dx} \\ $$$$=\int_{−\pi} ^{\:\pi} \left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right){dx}+\int_{−\pi} ^{\pi} \underset{−−−−−−−−{even}−−−−−−−−−} {\left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right)}\left(\underset{−−−−−−{odd}−−−−−} {{sinx}+…+{sin}\mathrm{13}{x}}\right){dx} \\ $$$$=\int_{−\pi} ^{\:\pi} \left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right){dx}+\mathrm{0} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\:\pi} \left(\mathrm{1}+{cosx}+{cos}\mathrm{2}{x}+….+{cos}\mathrm{13}{x}\right){dx} \\ $$$$=\mathrm{2}\left({x}+\mathrm{sin}\:{x}+\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{2}}+…+\frac{\mathrm{sin}\:\mathrm{13}{x}}{\mathrm{13}}\right)_{\mathrm{0}} ^{\pi} \\ $$$$=\mathrm{2}\pi \\ $$
Commented by rahul 19 last updated on 13/Feb/19
thank you sir!
Answered by mr W last updated on 13/Feb/19
$${Q}\left(\mathrm{4}\right) \\ $$$${let}\:{I}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\int_{\mathrm{0}} ^{\mathrm{2}} \left[\left({x}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1}\right]^{\frac{\mathrm{1}}{\mathrm{3}}} {d}\left({x}+\mathrm{1}\right) \\ $$$$=\int_{\mathrm{1}} ^{\mathrm{3}} \left({t}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dt} \\ $$$$=\int_{\mathrm{1}} ^{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\int_{\mathrm{1}} ^{\mathrm{3}} {g}\left({x}\right){dx} \\ $$$${with}\:{g}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$ \\ $$$$ \\ $$$${let}\:{f}\left({x}\right)=\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}={y} \\ $$$${x}=\left({y}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} ={g}\left({x}\right) \\ $$$$ \\ $$$${let}\:{I}_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}{dx}=\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx} \\ $$$${let}\:{x}={g}\left({t}\right) \\ $$$${f}\left({x}\right)={f}\left({g}\left({t}\right)\right)={t} \\ $$$${dx}={g}'\left({t}\right){dt} \\ $$$${I}_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}=\int_{{f}\left(\mathrm{0}\right)} ^{{f}\left(\mathrm{2}\right)} {tg}'\left({t}\right){dt}=\int_{\mathrm{1}} ^{\mathrm{3}} {tg}'\left({t}\right){dt} \\ $$$$=\int_{\mathrm{1}} ^{\mathrm{3}} {xg}'\left({x}\right){dx} \\ $$$$ \\ $$$${I}={I}_{\mathrm{1}} +{I}_{\mathrm{2}} =\int_{\mathrm{1}} ^{\mathrm{3}} {xg}'\left({x}\right){dx}+\int_{\mathrm{1}} ^{\mathrm{3}} {g}\left({x}\right){dx} \\ $$$$=\int_{\mathrm{1}} ^{\mathrm{3}} \left[{g}\left({x}\right)+{xg}'\left({x}\right)\right]{dx} \\ $$$$=\left[{xg}\left({x}\right)\right]_{\mathrm{1}} ^{\mathrm{3}} \\ $$$$=\mathrm{3}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}\left(\mathrm{1}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$=\mathrm{3}×\mathrm{8}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$=\mathrm{3}×\mathrm{2} \\ $$$$=\mathrm{6} \\ $$$$\Rightarrow\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\left(\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:+\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:\right){dx}\:=\mathrm{6} \\ $$
Commented by rahul 19 last updated on 14/Feb/19
$${Thanks}\:{sir}! \\ $$