The mathematical mood swing is real! The top integral (∫ 1/x^7 dx) evaluates to a negative constant (-1/6x^6), explaining the happy expression. But add just a +1 to the denominator, and suddenly you're dealing with ∫ 1/(x+1) dx, which gives you ln|x+1| - a logarithmic nightmare with no elementary antiderivative. No wonder the mood shifted from "I solved it!" to "I'm mathematically doomed." Calculus really can turn your smile upside down faster than you can say "integration by parts."
What A Difference A Constant Makes

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