Calculate the integral .
Evaluate the integral by following steps:
step0: Evaluate using formulas and rules:
step1: Evaluate the integral:
step2: Calculate the absolute value:
step3: Evaluate the power:
step4: Add the constant of integral C:
The integral of with respect to is , where is an arbitrary constant.
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The Deep Dive
This integral can be tackled by recognizing that suggests the use of a trigonometric substitution. Since , we can let . This simplifies the integral since . Consequently, you can express in terms of and subsequently update the limits as you transform the integral. This technique highlights the beauty and utility of trigonometric functions in simplifying complex integrals.
Another powerful approach is recognizing the integral as an opportunity to apply integration by parts if necessary. Though, in this case, a direct substitution yields a more seamless solution. Remembering to transform your variables accordingly, and paying careful attention to constant factors, helps avoid common pitfalls. It makes for a fun puzzle to navigate through different methods and observe how they lead to the same delightful answer in the end!