Final answer:
To evaluate the integral ∫(4 to 1) sin(√x) dx, a substitution method is employed, changing variables and limits, leading to the integration with respect to the new variable and computing the final value.
Step-by-step explanation:
The student asked to evaluate the integral of the function sin(√x) with respect to x from 1 to 4. This type of problem involves finding the area under the curve of the function sin(√x) between the given limits. To solve it, we can use a substitution method where we let u = √x, and therefore du = (1/(2√x)) dx, or dx = 2u du. Substituting this into the integral, we have to change the limits of integration to match the new variable u. Since u = √x, when x = 1, u = 1, and when x = 4, u = 2. We then integrate with respect to u and apply the new limits. Finally, we compute the integral to find the area.