This doesn't look very promising, and besides, it doesn't look like you did your substitution correctly. If ##u = 4 - r^2##, what is du? You can't simply replace dr by du.Arnoldjavs3 said:Homework Statement
$$\int_{0}^{2} r\sqrt{5-\sqrt{4-r^2}} dr$$
Homework Equations
The Attempt at a Solution
would i substitute ##u=4-r^2##?
After of which I would input into the integral and get:
$$\int_{0}^{2} \sqrt{5-\sqrt{u}}du$$
As I said, this doesn't seem very promising. I would try this substitution: ##u = \sqrt{4 - r^2}##, or equivalently, ##u^2 = 4 - r^2##.Arnoldjavs3 said:What would I do here? Do I just work inside the radical(so 5r - 2/3u^3/2)? Or do I have to account that it's inside a radical when solving?
The substitution rule for integrals, also known as the u-substitution method, is a technique used to simplify and solve integrals by making a substitution for the variable in the integrand.
The substitution rule should be used when the integrand contains a complicated function or when the integral involves a trigonometric function that can be simplified using a substitution.
To apply the substitution rule, you must first identify a substitution, u, that will simplify the integrand. Then, substitute u into the integral and replace all instances of the original variable with u. Finally, solve the integral with respect to u and substitute the original variable back in the answer.
The substitution rule is used when the integrand can be simplified by substituting a new variable, while integration by parts is used to integrate the product of two functions. The substitution rule is often used for integrals with complicated functions, while integration by parts is used for integrals with two functions.
Yes, the substitution rule can be used for definite integrals. When solving a definite integral using the substitution rule, the limits of integration must also be substituted with the new variable u.