Integration by Parts

Compute $\int xe^{5x}dx$ and $\displaystyle \int_{\pi/4}^{\pi/3} cos(x) ln(sin(x)) \, dx$

When approaching integration by parts, it's helpful to think of the method as a way to break down complex integrals into simpler parts. The technique comes from the product rule for differentiation, and it involves identifying parts of the integrand that can be differentiated easily and parts that can be integrated easily. The goal is to reduce the problem to a simpler form after performing the integration. In the case of the first integral, you'll identify one part of the expression to differentiate and another part to integrate. After applying this method once, you may need to apply it again to finish the problem.

For definite integrals involving trigonometric functions like cosine and natural logarithms, the process can involve using integration by parts or other methods to simplify the integral. You'll often look for a way to differentiate one part of the integrand and integrate another, with the goal of either reducing the complexity of the integrand or applying limits directly to evaluate the definite integral. In problems like these, it's important to keep track of both the algebraic manipulations and the limits of integration to ensure an accurate final answer.

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