bobsmith76 said:Homework Statement
The Attempt at a Solution
I don't see why the √v disappears in this step
I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^-1/2)/(1/2)
The purpose of substitution in a definite integral is to simplify the integrand and make it easier to evaluate. This is done by replacing the variable in the integrand with a new variable, and then using the appropriate substitution rule to rewrite the integral in terms of the new variable.
The substitution variable is typically chosen to be a function of the variable in the original integrand. It should also be chosen in a way that will simplify the integrand and make it easier to evaluate. Common choices for substitution variables include u, x^2, sin(x), or e^x.
The general process for substitution in a definite integral is as follows:
Yes, there are certain restrictions on when substitution can be used in a definite integral. The most important restriction is that the substitution must be a one-to-one function. This means that each value of the substitution variable must correspond to exactly one value of the original variable. Additionally, the limits of integration must also be adjusted to reflect the change in variables.
No, substitution cannot be used to solve all definite integrals. There are certain integrals that cannot be simplified using substitution, and may require other integration techniques such as integration by parts or trigonometric identities. It is important to carefully consider the integrand and choose the appropriate method for evaluation.