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PART III. This will be considered as your P1 Quiz No. 3:

Ask by Long Nunez. in the Philippines
Jan 29,2025

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Here are the solutions to the integrals:

Solución

Sure, let’s solve each of the integrals you’ve provided step by step.

Problem 1

Solution:
Let’s make the substitution:
Substituting into the integral:
Answer:

Problem 2

Solution:
Use the product-to-sum identities:
Now, integrate:
Both integrals of sine over a full period are zero:
Answer:

Problem 3

Solution:
Recall that , so:
Let’s set:
Rewriting the integral:
Answer:

Problem 4

Solution:
Express in terms of exponentials:
Substitute into the integral:
Answer:

Problem 5

Solution:
Recall the derivative of :
Thus,
Answer:

Problem 6

Solution:
Notice that:
and
Thus,
Applying the limits:
Evaluate :
So,
Thus,
Answer:

Summary of All Solutions

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Beyond the Answer

Evaluating integrals can feel daunting, but knowing a bit of their history can help. Many of the techniques we use today, like integration by parts or substitution, were developed by mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Their revolutionary approaches laid the groundwork for modern calculus, turning complex functions into more manageable parts and allowing us to tackle a wide array of problems in physics, engineering, and beyond!
When solving integrals, it’s easy to fall into common pitfalls. For instance, when integrating functions involving logarithms, many forget to adjust the limits of integration if they are applying a substitution. Another frequent mistake is neglecting to identify the appropriate formula or technique for integration, like overlooking trigonometric identities for products of sine and cosine. Keep an eye out for these, and practice will make you a master of integrals before you know it!

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