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challenge integral

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challenge integral

Evaluate integral e^(cos x) dx

Also what is integral (x^2 + 1)^n dx , where n is a constant

Evaluate them as indefinite integrals (no limits)

You can't. Use numerical analysis to obtain an approximation (Try using Taylor's expansion, so you may truncate the series where you want, with a lot of work BTW). An exact solution, in terms of elementary functions, is not known for such integral. For the second one use a recursive formula (or expand out the binomial).
Just only a few classes of functions are integrable in closed form, not that we want.

What are the limits of integration? Is this meant as
definite or indefinite integral?

The second integral can be expanded by binomial theorem and integrated term by term, if n is an integer.

If n is not an integer you will get an infinite series. For the first you can use the power series for e^u and integrate the powers of cosx.

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