u(1)=(1)5 +2(1) = 3 not 2 .jtt said:Homework Statement
make a u- substitution and integrate from u(a) to u(b)
Homework Equations
∫[0,1] √(t^5+2t) (5t^4+2) dt
The Attempt at a Solution
u= t^5+2t du= 5t^4+2
u(1)=2 u(0)=0
∫[0,1] √(u) du
(2/3)(u)^3/2+c l[0,2]
(2/3)( 0^5+2(0))^3/2- (2/3)(2^5+2(2))^3/2
0+ 24^3/2
the textbook answer is 2√3 but I don't know where I messed up
U substitution, also known as the substitution method, is a technique used to simplify an integral by replacing the variable of integration with a new variable, often denoted as u. This allows for easier integration and evaluation of the integral.
U substitution is most commonly used when the integrand (the expression inside the integral) contains a function within a function, such as sin(x^2) or e^(x^3). It is also useful when the integrand contains a product of functions, or when the limits of integration are in terms of the variable being substituted.
The u chosen for substitution should be based on the derivative of the function inside the integral. In most cases, u will be a simpler function than the original integrand, making integration easier. Additionally, u should be chosen such that the differential du (dx or dy) can be easily substituted in the integral.
Yes, U substitution can be used for definite integrals with multiple variables, as long as the variable being substituted appears in both the limits of integration and the integrand. However, in these cases, the expression for u may be more complicated and may require further algebraic manipulation.
One common mistake is forgetting to change the limits of integration from the original variable to u. It is important to substitute both the integrand and the limits of integration with the new variable. Additionally, it is important to check the signs of the limits when using u substitution, as they may change depending on the direction of the substitution.