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∫e−x+sin⁡xe−x+cos⁡x dx\int \frac{e^{-x} + \sin x}{e^{-x} + \cos x} \, dx

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Q. ∫e−x+sin⁡xe−x+cos⁡x dx\int \frac{e^{-x} + \sin x}{e^{-x} + \cos x} \, dx
  1. Simplify Integral: Simplify the integral if possible.\newlineWe have the integral:\newline∫(e−x+sin⁡(x))/(e−x+cos⁡(x)) dx\int(\mathrm{e}^{-x} + \sin(x)) / (\mathrm{e}^{-x} + \cos(x)) \, dx\newlineThis integral does not simplify easily due to the presence of both exponential and trigonometric functions in the numerator and denominator. There is no obvious substitution or simplification that can be made.
  2. Substitution Attempt: Look for a substitution that can simplify the integral.\newlineLet's try the substitution u=e−x+cos⁡(x)u = e^{-x} + \cos(x), which means du=(−e−x−sin⁡(x))dxdu = (-e^{-x} - \sin(x)) dx. However, this substitution does not seem to help because the numerator is e−x+sin⁡(x)e^{-x} + \sin(x), not −e−x−sin⁡(x)-e^{-x} - \sin(x). Therefore, this substitution will not simplify the integral.
  3. Consider Alternative Methods: Consider alternative methods. Since substitution does not seem to work, we might consider other methods such as integration by parts or partial fractions. However, these methods do not apply here because we do not have a product of functions in the integrand, and the denominator is not a polynomial.
  4. Cancel Terms: Look for a way to cancel terms.\newlineWe can try to add and subtract e−xe^{-x} in the numerator to create a term that will cancel with the denominator:\newline∫e−x+sin⁡(x)e−x+cos⁡(x) dx\int\frac{e^{-x} + \sin(x)}{e^{-x} + \cos(x)} \, dx\newline= ∫(e−x+cos⁡(x))+(sin⁡(x)−cos⁡(x))e−x+cos⁡(x) dx\int\frac{(e^{-x} + \cos(x)) + (\sin(x) - \cos(x))}{e^{-x} + \cos(x)} \, dx\newline= ∫(1+sin⁡(x)−cos⁡(x)e−x+cos⁡(x))dx\int\left(1 + \frac{\sin(x) - \cos(x)}{e^{-x} + \cos(x)}\right) dx\newlineNow we have separated the integral into two parts:\newline∫1 dx+∫sin⁡(x)−cos⁡(x)e−x+cos⁡(x) dx\int 1 \, dx + \int\frac{\sin(x) - \cos(x)}{e^{-x} + \cos(x)} \, dx
  5. Integrate First Part: Integrate the first part.\newlineThe integral of 11 with respect to xx is simply xx. So we have:\newline∫(1) dx=x\int(1) \, dx = x
  6. Consider Second Part: Consider the second part of the integral.\newlineThe second part of the integral is more complex:\newline∫(sin⁡(x)−cos⁡(x)e−x+cos⁡(x))dx\int\left(\frac{\sin(x) - \cos(x)}{e^{-x} + \cos(x)}\right) dx\newlineThis does not have an obvious antiderivative, and standard methods do not apply. This suggests that the integral may not have a closed-form solution expressible in terms of elementary functions.
  7. Conclude Evaluation: Conclude the evaluation of the integral.\newlineSince the second part of the integral does not have a standard antiderivative, we cannot proceed further with elementary methods. Therefore, we conclude that the integral of (e−x+sin⁡(x))/(e−x+cos⁡(x))(e^{-x} + \sin(x)) / (e^{-x} + \cos(x)) with respect to xx does not have a closed-form solution in terms of elementary functions.

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