0-pi-4-tsint-1-cos-2-t-dt-0-pi-4-tcost-1-sin-2-t-dt-true-or-false-

Question Number 149023 by ArielVyny last updated on 02/Aug/21

∫_0 ^(π/4) ((tsint)/(1+cos^2 t))dt=∫_0 ^(π/4) ((tcost)/(1+sin^2 t))dt true or false ??

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{tsint}}{\mathrm{1}+{cos}^{\mathrm{2}} {t}}{dt}=\overset{\frac{\pi}{\mathrm{4}}} {\int}_{\mathrm{0}} \frac{{tcost}}{\mathrm{1}+{sin}^{\mathrm{2}} {t}}{dt} \\ $$$${true}\:{or}\:{false}\:?? \\ $$

Answered by mindispower last updated on 02/Aug/21

((sin(t))/(1+cos^2 (t)))<((cos(t))/(1+sin^2 (t))),∀t∈[0,(π/4)[ proff cos(t)>sin(t)≥0⇒ (1/(1+cos^2 (t)))<(1/(1+sin^2 (t)))⇒((sin(t))/(1+cos^2 (t)))<((cos(t))/(1+sin^2 (t))) ⇒0≤t((sin(t))/(1+cos^2 (t)))<((tcos(t))/(1+sin^2 (t))) ∫_0 ^(π/4) ((tsin(t))/(1+cos^2 (t)))dt<∫_0 ^(π/4) ((tcos(t))/(1+sin^2 (t)))dt

$$\frac{{sin}\left({t}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({t}\right)}<\frac{{cos}\left({t}\right)}{\mathrm{1}+{sin}^{\mathrm{2}} \left({t}\right)},\forall{t}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\left[\right.\right. \\ $$$${proff}\:\:{cos}\left({t}\right)>{sin}\left({t}\right)\geqslant\mathrm{0}\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({t}\right)}<\frac{\mathrm{1}}{\mathrm{1}+{sin}^{\mathrm{2}} \left({t}\right)}\Rightarrow\frac{{sin}\left({t}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({t}\right)}<\frac{{cos}\left({t}\right)}{\mathrm{1}+{sin}^{\mathrm{2}} \left({t}\right)} \\ $$$$\Rightarrow\mathrm{0}\leqslant{t}\frac{{sin}\left({t}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({t}\right)}<\frac{{tcos}\left({t}\right)}{\mathrm{1}+{sin}^{\mathrm{2}} \left({t}\right)} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{tsin}\left({t}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({t}\right)}{dt}<\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{tcos}\left({t}\right)}{\mathrm{1}+{sin}^{\mathrm{2}} \left({t}\right)}{dt} \\ $$$$ \\ $$

Commented by mindispower last updated on 03/Aug/21