SteamKing said:
It ultimately depends on the context of the problem and the method being used to solve it. Generally, we jump to a different integral when it is easier to solve or when it will provide a more efficient solution.
The thought process involves looking for patterns and similarities between the two integrals. We may also use algebraic manipulations or substitution techniques to transform one integral into the other, making it easier to solve.
No, it is not always necessary. Sometimes, we can solve the original integral without jumping to a different one. However, in some cases, jumping to a different integral can make the solution more efficient or even possible.
This also depends on the context and the integrals involved. Some common methods include u-substitution, integration by parts, and trigonometric identities. Choosing the right method often involves trial and error and experience.
Practice and familiarity with different integration techniques can be helpful in determining when and how to jump to a different integral. It is also important to pay attention to the structure and patterns of the integrals to make the process easier.
