91Ó°ÊÓ

Integration by parts is a technique derived from the product rule of differentiation. It's a clever method when dealing with integrals involving the product of two functions. The main formula used here is \( \int u \, dv = uv - \int v \, du \). However, the key is selecting the right parts for \( u \) and \( dv \). In this problem, we identified \( u = \text{arctan}(u) \) and \( dv = 2u \, du \), which are strategic choices to ease the integration process.

• Selecting \( u \): Choose the part of the integral that becomes simpler when differentiated. In our case, \( \text{arctan}(u) \) simplifies to \( \frac{1}{1+u^2} \).
• Selecting \( dv \): Choose the part that is easily integrable. Here, \( 2u \, du \) integrates to \( u^2 \).

This method reduces the complexity of the original integral by breaking it down into simpler parts. After applying the integration by parts formula, careful simplification is needed to solve the final integral component for a complete solution.

The substitution method is a powerful tool in integration, especially when faced with inverse functions or composite expressions. In this exercise, we substitute \( u = \sqrt{x} \), converting the original integral into a more manageable form. This allows us to redirect our focus on variable \( u \), simplifying successive calculations.

• New variables: By setting \( u = \sqrt{x} \), we find \( x = u^2 \) and consequently \( dx = 2u \, du \).
• Transformation: The integral \( \int \arctan(\sqrt{x}) \, dx \) is transformed into \( \int \arctan(u) \cdot 2u \, du \), easing the process of integration.

This method leverages the idea of simplifying the integrand function, making the subsequent steps, like applying integration by parts, more straightforward and reducing the risk of errors.

Differential substitution is crucial in accurately transforming and solving an integral. After defining a substitution for the key variable, we must also update the differential \( dx \) accordingly. In this exercise, after setting \( u = \sqrt{x} \), we derive that \( dx = 2u \, du \), which perfectly complements our substitution.

• Calculating \( dx \): Upon substitution, calculate the differential by differentiating \( x = u^2 \), resulting in \( dx = 2u \, du \).
• Completing transformation: Replace both the function and differential in the integrand, yielding \( \int \arctan(u) \cdot 2u \, du \).

Accurate differential substitution ensures the new integral reflects the true nature of the original, allowing further techniques like integration by parts to be applied effectively. Back-substitution at the end restores the solution to the original variable, \( x \), bringing clarity and harmony to the problem-solving process.