91Ó°ÊÓ

Integration by substitution is a technique used in calculus to simplify the process of evaluating integrals. It involves changing the variable of integration to a new variable, which can make the integral easier to solve. This method is particularly useful when you have a composite function that is difficult to integrate directly.

In this exercise, the original integral is changed by selecting a suitable substitution. Here, the substitution chosen is \(u = x^3 + 1\). This choice transforms the function \( \sqrt{x^3 + 1} \) into \( \sqrt{u} \), making the integral less complex.

After substituting, the differential \(dx\) also needs to be expressed in terms of \(du\). We find \(dx\) by differentiating the expression for \(u\), leading to \(du = 3x^2 dx\), hence \(dx = \frac{du}{3x^2}\). This transformation dramatically simplifies the integral, allowing us to proceed easily with integration.

When performing integration by substitution, it's crucial to also adjust the limits of integration to match the new variable. This ensures that the final result is calculated over the correct interval.

For our substitution \(u = x^3 + 1\), the original limits were from 1 to 2 with respect to \(x\). By substituting our new variable, the limits translate to \(u = 2\) when \(x = 1\) and \(u = 9\) when \(x = 2\) because \(2^3 + 1 = 9\). Thus, the limits of integration change from \([1,2]\) to \([2,9]\).

This adjustment must always be done whenever you change the variable to ensure correct evaluation of the definite integral in terms of the new variable.

Solving calculus problems like this integral often involves several techniques and steps. It's not only about calculating integrals but applying problem-solving strategies effectively.

First, identify potential substitutions that simplify the function. Then translate the entire integral, including \(dx\) and the limits of integration, into the new variable framework. As demonstrated, each step requires careful calculation and verification to avoid mistakes.

Consider invariants like consistent integration limits and maintaining a clear transformation path. Doing so ensures that the solution aligns with the original problem constraints and gives insight into calculus problem-solving strategies.

Once the integral has been expressed in its simplified form, it's crucial to evaluate it correctly. After performing the substitution, integration becomes straightforward.

For \(u^{1/2}\), the antiderivative is \( (2/3)u^{3/2} \), multiplied by any constants outside the integral. Here, this is multiplied by \(2/3\) due to the initial integral setup, resulting in \( (4/9)u^{3/2} \).

Finally, evaluate the definite integral by substituting the new limits of integration. Insert the upper limit first and then the lower one, subtracting the two results to find the final value. For this example, it yields \((4/9) \times (27 - 2.82) = 10.86\), showing the importance of careful calculation throughout the process.