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Evaluate-0-pi-2-sin-2x-e-cos-2-x-dx-




Question Number 14395 by tawa tawa last updated on 31/May/17
Evaluate      ∫_(   0) ^(  (π/2))  sin(2x) e^(cos^2 (x))  dx
$$\mathrm{Evaluate}\:\:\:\:\:\:\int_{\:\:\:\mathrm{0}} ^{\:\:\frac{\pi}{\mathrm{2}}} \:\mathrm{sin}\left(\mathrm{2x}\right)\:\mathrm{e}^{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)} \:\mathrm{dx}\: \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 31/May/17
u=cos^2 x⇒du=−2sinx.cosxdx=−sin2xdx  I=∫−e^u du=−e^u +C=−e^(cos^2 x) +C.  I=F((π/2))−F(0)=−e^0 +e^1 =e−1 .■
$${u}={cos}^{\mathrm{2}} {x}\Rightarrow{du}=−\mathrm{2}{sinx}.{cosxdx}=−{sin}\mathrm{2}{xdx} \\ $$$${I}=\int−{e}^{{u}} {du}=−{e}^{{u}} +{C}=−{e}^{{cos}^{\mathrm{2}} {x}} +{C}. \\ $$$${I}={F}\left(\frac{\pi}{\mathrm{2}}\right)−{F}\left(\mathrm{0}\right)=−{e}^{\mathrm{0}} +{e}^{\mathrm{1}} ={e}−\mathrm{1}\:.\blacksquare \\ $$
Commented by tawa tawa last updated on 31/May/17
God bless you sir.
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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