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S-0-sin-e-x-cos-e-x-dx-1-Solve-S-Also-Does-2-0-sin-e-x-cos-e-x-dx-




Question Number 3988 by Filup last updated on 26/Dec/15
S=∫_(−∞) ^( 0) [sin(e^x ) cos(e^(−x) )]dx    (1)   Solve S    Also, Does:  (2)   ∫_0 ^( ∞) [sin(e^x ) cos(e^(−x) )]dx=∞?
$${S}=\int_{−\infty} ^{\:\mathrm{0}} \left[\mathrm{sin}\left({e}^{{x}} \right)\:\mathrm{cos}\left({e}^{−{x}} \right)\right]{dx} \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\:\:\mathrm{Solve}\:{S} \\ $$$$ \\ $$$$\mathrm{Also},\:\mathrm{Does}: \\ $$$$\left(\mathrm{2}\right)\:\:\:\int_{\mathrm{0}} ^{\:\infty} \left[\mathrm{sin}\left({e}^{{x}} \right)\:\mathrm{cos}\left({e}^{−{x}} \right)\right]{dx}=\infty? \\ $$
Commented by Yozzii last updated on 26/Dec/15
sinpcosq=(1/2)(sin(p+q)+sin(p−q))  ⇒S=(1/2)∫_(−∞) ^0 [sin(e^x +e^(−x) )+sin(e^x −e^(−x) )]dx  ∫sin(2coshx)dx=?,∫sin(2sinhx)dx=?
$${sinpcosq}=\frac{\mathrm{1}}{\mathrm{2}}\left({sin}\left({p}+{q}\right)+{sin}\left({p}−{q}\right)\right) \\ $$$$\Rightarrow{S}=\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\mathrm{0}} \left[{sin}\left({e}^{{x}} +{e}^{−{x}} \right)+{sin}\left({e}^{{x}} −{e}^{−{x}} \right)\right]{dx} \\ $$$$\int{sin}\left(\mathrm{2}{coshx}\right){dx}=?,\int{sin}\left(\mathrm{2}{sinhx}\right){dx}=? \\ $$
Commented by Filup last updated on 26/Dec/15

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