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=∞?](Q3988.png)
$${S}=\int_{−\infty} ^{\:\mathrm{0}} \left[\mathrm{sin}\left({e}^{{x}} \right)\:\mathrm{cos}\left({e}^{−{x}} \right)\right]{dx} \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\:\:\mathrm{Solve}\:{S} \\ $$$$ \\ $$$$\mathrm{Also},\:\mathrm{Does}: \\ $$$$\left(\mathrm{2}\right)\:\:\:\int_{\mathrm{0}} ^{\:\infty} \left[\mathrm{sin}\left({e}^{{x}} \right)\:\mathrm{cos}\left({e}^{−{x}} \right)\right]{dx}=\infty? \\ $$
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=?](Q4008.png)
$${sinpcosq}=\frac{\mathrm{1}}{\mathrm{2}}\left({sin}\left({p}+{q}\right)+{sin}\left({p}−{q}\right)\right) \\ $$$$\Rightarrow{S}=\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\mathrm{0}} \left[{sin}\left({e}^{{x}} +{e}^{−{x}} \right)+{sin}\left({e}^{{x}} −{e}^{−{x}} \right)\right]{dx} \\ $$$$\int{sin}\left(\mathrm{2}{coshx}\right){dx}=?,\int{sin}\left(\mathrm{2}{sinhx}\right){dx}=? \\ $$
Commented by Filup last updated on 26/Dec/15
