Question

Use a change of variables to evaluate the following definite integral. Determine a change of variables from x to u.

Answer 1

The definite integra l
`\(\int_(1)^(2) (4x - 3) \,dx\`) evaluates to 5 after a change of variables, with \(u = 4x - 3\).

Step-by-step explanation:

To solve the definite integral ∫(4x - 3)dx from 1 to 2, we employ a change of variables. Let \(u = 4x - 3\), and differentiating both sides with respect to x yields du/dx = 4. Solving for dx, we find dx = du/4. Substituting these expressions into the integral and adjusting the limits accordingly, we obtain
`\(\int_(1)^(2) (4x - 3) \,dx = \int_(1)^(5) u \cdot (1)/(4) \,du\)`. Integrating with respect to u, the result is

`\((1)/(4) \left[(u^2)/(2)\right]_(1)^(5) = (1)/(4) \left((25)/(2) - (1)/(2)\right) = 5\)`.

Thus, the original integral is equivalent to 5 after the change of variables.

This substitution simplifies the integral, making it more manageable. The transformation from x to u streamlines the integration process and allows us to evaluate the integral more easily in terms of the new variable. The concept of changing variables is a fundamental technique in calculus, frequently applied to simplify integrals and uncover more straightforward solutions. In this case, the substitution
`\(u = 4x - 3\)` proves effective in transforming the original integral into a more accessible form, ultimately leading to the final answer of 5.

In conclusion, the change of variables \(u = 4x - 3\) not only simplifies the given definite integral but also showcases the versatility and utility of this technique in calculus problem-solving. This approach facilitates the integration process, making it more amenable to calculation and providing a clear path to the final result of 5.