The discussion focuses on the application of u-substitution in definite integrals, specifically integrating the function ∫[0,1] √(t^5+2t) (5t^4+2) dt. The correct substitution is u = t^5 + 2t, leading to du = (5t^4 + 2) dt. The limits of integration change from t=0 to t=1, resulting in u(0)=0 and u(1)=3. The final integral simplifies to ∫[0,3] √(u) du, which evaluates to 2√3, confirming the textbook answer.
PREREQUISITESStudents studying calculus, particularly those learning integration techniques, and educators seeking to clarify u-substitution methods in definite integrals.
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