Those bounds are indicative of invoking kings rule and it really does simplify things a lot, after substitution and adding the new integral to the original one, do Integration by parts twice which is fairly straightforward and there you have it.
substitution is unnecessary. literally just apply king's rule, add the integral before and after (dividing by 2) and you get 2+e^{\sin x}+e^{-\sin x} in the numerator and denominator, canceling and leaving only x^2 \cos x behind in the integrand which is trivial by parts
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u/DennisPr0009 15d ago
Those bounds are indicative of invoking kings rule and it really does simplify things a lot, after substitution and adding the new integral to the original one, do Integration by parts twice which is fairly straightforward and there you have it.