d)

To tackle the integral , we can use integration by parts or a substitution method. If we let , then or equivalently, . Substituting this into the integral, we have:

This integral might seem tricky at first, but it can be solved using integration by parts. We can set and and find the relationship to get a formula that relates back to itself!

Solving integrals with logarithmic functions often reveals patterns and deeper connections in calculus, leading to unexpected and beautiful results. This method shows just how versatile calculus can be, going from simple powers of to more complex logarithmic transformations. So, wield your mathematical skills like a wizard and see how many magical results you can uncover with similar integrals!

If you’re looking for real-world applications, the integration of functions such as pops up in fields like signal processing where transforming signals into frequency components is essential. The logarithm relates to how we perceive sound intensity, while the cosine helps maintain oscillatory behavior, making it super relevant in engineering and physics!