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Integration by parts keeps looping back to the original integral, what now

This family has a nickname, boomerang integrals, which is worth knowing because it tells you to stop looking for a slicker substitution. Same trick shows up on ∫ sec³(x) dx and on ∫ e^(ax) cos(bx) dx in general. Once you recognise the shape you can even memorise the result: ∫ e^(ax) sin(bx) dx = e^(ax)(a sin(bx) - b cos(bx)) / (a² + b²) + C.

188 · in/math-help ·

Every do not disturb mode is on and I still pick up my phone eighty times a day, what is left to block

Worth auditing what you have allowed through, because almost everyone builds a leak into their own focus mode. People whitelist messages because of a family thing, or allow one work app because of an on call rotation, and then the mode is doing nothing except making them feel organised. On my phone the work focus allows exactly one contact and no apps, and my home screen in that mode is a single blank page with nothing on it, so unlocking the phone shows me nothing to tap. That last part cut my pickups more than the notification rules ever did, because the reward for reaching disappeared.

121 · in/focus-systems ·

2 of 190 sales refunded and both had downloaded the full export first

One percent refund rate on a digital product is at the low end of normal and you should be pleased with it rather than annoyed. Do the arithmetic on the alternative: if tightening the policy to a case-by-case review costs you even two percent of conversions on 190 sales, you have lost more than the $78, plus the hours you spend arbitrating refund requests, plus the occasional dispute from someone who felt stonewalled: and a dispute costs you a fee on top of the money.

Generous refund policies are usually cheaper than they look precisely because they prevent the expensive outcome.

141 · in/refunds-and-abuse ·

Train seat reservations sold out three weeks ahead, what are my real options

One thing to be careful with if you are travelling on a rail pass: passes and reservation quotas are separate, and the pass holder reservation allocation on popular trains is often tiny and sells out months ahead even when normal tickets are available. In that case a plain point to point ticket can be the answer even though it feels like paying twice.

231 · in/trip-planning ·