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Integrate cosh⁡(e3x)\cosh(e^{3x}) over the interval of −π-\pi to π\pi

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Full solution

Q. Integrate cosh⁡(e3x)\cosh(e^{3x}) over the interval of −π-\pi to π\pi
  1. Recognize Problem: Recognize that the function cosh⁡(e3x)\cosh(e^{3x}) is not a standard integral form, so we need to find a substitution or another method to integrate it.
  2. Substitution Attempt: Let's try a substitution. Let u=e3xu = e^{3x}, then du=3e3xdxdu = 3e^{3x}dx. We need to adjust for the 33, so dx=du3e3xdx = \frac{du}{3e^{3x}}.
  3. Integrate with Substitution: Substitute uu and dxdx into the integral. The new integral is 13∫cosh⁡(u)⋅(1u)du\frac{1}{3} \int \cosh(u) \cdot \left(\frac{1}{u}\right) du, with the limits of integration changing to e(−3π)e^{(-3\pi)} and e(3π)e^{(3\pi)} after substituting x=−πx = -\pi and x=πx = \pi.

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