Question Number 36415 by abdo.msup.com last updated on 01/Jun/18
$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({x}^{\mathrm{3}} \:+{x}\right){cos}^{\mathrm{2}} {xdx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\left({x}^{\mathrm{3}} \:+{x}\right){sin}^{\mathrm{2}} {xdx} \\ $$$${cslculate}\:{I}\:{and}\:{J}\:. \\ $$
Commented by abdo.msup.com last updated on 04/Jun/18
![we have I +J = ∫_0 ^(π/2) (x^3 +x)dx =[(x^4 /4) +(x^2 /2)]_0 ^(π/2) = (π^4 /(4.2^4 )) +(π^2 /(2.2^2 )) =(π^4 /(64)) +(π^2 /8) also we have I −J = ∫_0 ^(π/2) (x^3 +x)cos(2x)dx =_(2x=t) ∫_0 ^π { ((t/2))^3 +(t/2)}cos(t) (dt/2) =(1/2) ∫_0 ^π { (t^3 /8) +(t/2)}cos(t)dt = (1/(16)) ∫_0 ^π { t^3 +4t)cos(2t)dt by parts ∫_0 ^π { t^3 +4t)cos(2t)=[(1/2)(t^3 +4t)sin(2t)]_0 ^π −(1/2)∫_0 ^π {3t^2 +4}sin(2t)dt =−(1/2){ [ −(1/2)(3t^2 +4)cos(2t)]_0 ^π −∫_0 ^π −(1/2)(6t) cos(2t)dt =(1/4){( 3π^2 +4)−4} −(3/2)∫_0 ^π t cos(2t)dt =(3/4)π^2 −(3/2){ [(1/2)t sin(2t)]_0 ^π −∫_0 ^π (1/2)sin(2t)dt} =((3π^2 )/4) +(3/4) ∫_0 ^π sin(2t)dt =((3π^2 )/4) −(3/8)[cos(2t)]_0 ^π = ((3π^2 )/4) so I +J = (π^4 /(64)) +(π^2 /8) and I −J = ((3π^2 )/4) ⇒ 2I = (π^4 /(64)) + ((7π^2 )/8) ⇒ I = (π^4 /(128)) +((7π^2 )/(16)) and 2J =(π^4 /(64)) +(π^2 /8) −((3π^2 )/4) =(π^4 /(64)) −((5π^2 )/8) ⇒ J = (π^4 /(128)) −((5π^2 )/(16)) .](Q36729.png)
$${we}\:{have}\:{I}\:\:+{J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({x}^{\mathrm{3}} \:+{x}\right){dx} \\ $$$$=\left[\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\:\frac{\pi^{\mathrm{4}} }{\mathrm{4}.\mathrm{2}^{\mathrm{4}} }\:+\frac{\pi^{\mathrm{2}} }{\mathrm{2}.\mathrm{2}^{\mathrm{2}} }\:=\frac{\pi^{\mathrm{4}} }{\mathrm{64}}\:+\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\ $$$${also}\:{we}\:{have}\:{I}\:−{J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({x}^{\mathrm{3}} \:+{x}\right){cos}\left(\mathrm{2}{x}\right){dx} \\ $$$$=_{\mathrm{2}{x}={t}} \:\:\int_{\mathrm{0}} ^{\pi} \:\left\{\:\left(\frac{{t}}{\mathrm{2}}\right)^{\mathrm{3}} \:+\frac{{t}}{\mathrm{2}}\right\}{cos}\left({t}\right)\:\frac{{dt}}{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\pi} \:\:\left\{\:\frac{{t}^{\mathrm{3}} }{\mathrm{8}}\:+\frac{{t}}{\mathrm{2}}\right\}{cos}\left({t}\right){dt} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{16}}\:\int_{\mathrm{0}} ^{\pi} \:\left\{\:{t}^{\mathrm{3}} \:+\mathrm{4}{t}\right){cos}\left(\mathrm{2}{t}\right){dt}\:\:{by}\:{parts} \\ $$$$\int_{\mathrm{0}} ^{\pi} \:\left\{\:{t}^{\mathrm{3}} \:+\mathrm{4}{t}\right){cos}\left(\mathrm{2}{t}\right)=\left[\frac{\mathrm{1}}{\mathrm{2}}\left({t}^{\mathrm{3}} \:+\mathrm{4}{t}\right){sin}\left(\mathrm{2}{t}\right)\right]_{\mathrm{0}} ^{\pi} \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \:\left\{\mathrm{3}{t}^{\mathrm{2}} \:+\mathrm{4}\right\}{sin}\left(\mathrm{2}{t}\right){dt} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\left[\:−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{3}{t}^{\mathrm{2}} \:+\mathrm{4}\right){cos}\left(\mathrm{2}{t}\right)\right]_{\mathrm{0}} ^{\pi} \right. \\ $$$$−\int_{\mathrm{0}} ^{\pi} −\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{6}{t}\right)\:{cos}\left(\mathrm{2}{t}\right){dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left\{\left(\:\mathrm{3}\pi^{\mathrm{2}} +\mathrm{4}\right)−\mathrm{4}\right\}\:−\frac{\mathrm{3}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \:{t}\:{cos}\left(\mathrm{2}{t}\right){dt} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}\pi^{\mathrm{2}} \:\:−\frac{\mathrm{3}}{\mathrm{2}}\left\{\:\:\left[\frac{\mathrm{1}}{\mathrm{2}}{t}\:{sin}\left(\mathrm{2}{t}\right)\right]_{\mathrm{0}} ^{\pi} \:−\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{1}}{\mathrm{2}}{sin}\left(\mathrm{2}{t}\right){dt}\right\} \\ $$$$=\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{4}}\:\:+\frac{\mathrm{3}}{\mathrm{4}}\:\int_{\mathrm{0}} ^{\pi} \:{sin}\left(\mathrm{2}{t}\right){dt} \\ $$$$=\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{4}}\:\:−\frac{\mathrm{3}}{\mathrm{8}}\left[{cos}\left(\mathrm{2}{t}\right)\right]_{\mathrm{0}} ^{\pi} \:=\:\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{4}}\:{so} \\ $$$${I}\:+{J}\:=\:\frac{\pi^{\mathrm{4}} }{\mathrm{64}}\:+\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:\:{and}\:{I}\:−{J}\:=\:\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{4}}\:\Rightarrow \\ $$$$\mathrm{2}{I}\:=\:\frac{\pi^{\mathrm{4}} }{\mathrm{64}}\:+\:\frac{\mathrm{7}\pi^{\mathrm{2}} }{\mathrm{8}}\:\Rightarrow\:{I}\:=\:\frac{\pi^{\mathrm{4}} }{\mathrm{128}}\:+\frac{\mathrm{7}\pi^{\mathrm{2}} }{\mathrm{16}}\:\:{and} \\ $$$$\mathrm{2}{J}\:=\frac{\pi^{\mathrm{4}} }{\mathrm{64}}\:+\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:−\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{4}}\:=\frac{\pi^{\mathrm{4}} }{\mathrm{64}}\:−\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{8}}\:\Rightarrow \\ $$$${J}\:=\:\frac{\pi^{\mathrm{4}} }{\mathrm{128}}\:−\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{16}}\:. \\ $$