Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx

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Recognize that the integral is of the form \(\int e^x \sin(e^x) \, dx\), where the argument of the sine function is \(e^x\) and the outside factor is also \(e^x\).

Use substitution by letting \(u = e^x\). Then, compute the differential \(du = e^x \, dx\), which implies \(dx = \frac{du}{e^x} = \frac{du}{u}\).

Rewrite the integral in terms of \(u\): since \(e^x \, dx = du\), the integral becomes \(\int \sin(u) \, du\).

Integrate \(\sin(u)\) with respect to \(u\), which is \(-\cos(u) + C\).

Substitute back \(u = e^x\) to express the answer in terms of \(x\): the integral is \(-\cos(e^x) + C\).

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