Is there a hand-written way of integrating sec^2(x)cos^8(x) USING calculus one methods? No product rule.

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Recall that \[sec\ x = \frac{1}{cos\ x}\]
So you can just cancel 2 of the cosines.

Yes but they're to different powers.

Yes but
\[x^8 * \frac{1}{x^2} = x^6\]

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