calculus
posted by adela .
using the fundamental theorem of calculus what is the derivative of the definite integral from x^3 to sqrt x of (sqrt t) sin t dt

The first fundamental theorem of calculus states that if f(x) is continuous, real and defined on [a,b], and
F(x)=∫f(x)dx from a to b
then
F(x) is continuous on [a,b] and differentiable on (a,b), then
F'(x) = f(x).
In this case f(t)=sqrt(t)sin(t), definite integral is calculated from x³ to √(x).
Thus if the above theorem to apply, t must be nonnegative, which implies that x>0.
If the condition is satisfied, then f(t) is continuous and defined on [0,∞], and consequently F'(t) = f(t). 
Yoto
Respond to this Question
Similar Questions

Calculus  Second Order Differential Equations
Posted by COFFEE on Monday, July 9, 2007 at 9:10pm. download mp3 free instrumental remix Solve the initialvalue problem. y'' + 4y' + 6y = 0 , y(0) = 2 , y'(0) = 4 r^2+4r+6=0, r=(16 +/ Sqrt(4^24(1)(6)))/2(1) r=(16 +/ Sqrt(8)) r=8 … 
Calculus
Please look at my work below: Solve the initialvalue problem. y'' + 4y' + 6y = 0 , y(0) = 2 , y'(0) = 4 r^2+4r+6=0, r=(16 +/ Sqrt(4^24(1)(6)))/2(1) r=(16 +/ Sqrt(8)) r=8 +/ Sqrt(2)*i, alpha=8, Beta=Sqrt(2) y(0)=2, e^(8*0)*(c1*cos(0)+c2*sin(0))=c2=2 … 
Math/Calculus
Solve the initialvalue problem. Am I using the wrong value for beta here, 2sqrt(2) or am I making a mistake somewhere else? 
calculus
Calculate definite integral of dx/(x^4 * sqrt(x^2 + 3)) Over (1,3) I start with the substitution x = sqrt(3)*tan t so: sqrt(x^2 + 3) = sqrt(3) * sec t dx = sqrt(3) * sec^2 t dt x^4 = 9 * tan^4 t The integral simplifies to: = dt/(tan^3 … 
Maths Calculus Derivative Integral  Urgent Please
Use 2nd Fundamental Theorem of Calculus to find derivative of f(x) = integral of 2x^2 (at the top) to x5 (at the bottom) of square root of Sin(x)dx 
Calculus
"Leave the answer as a definite integral, but indicate how it could by evaluated by using the fundamental theorem of calculus." I solved the problem to a definite integral. Proceeding via the fundamental theorem, would involve finding … 
ap calculus
Which of the following definite integrals gives the length of y = e^(e^x) between x=0 and x=1? 
Calculus Fundamental Theorem
Evaluate the definite integral. function: (t+8)(t^2+3) with respect to variable t lower limit: sqrt(2) upper limit: sqrt(2) 
Calculus 2 (Differential Equation)
How would you solve the following problem explicitly? 
Calculus1
find the derivative using fundamental theorem of calculus integral of sin^3tdt from e^x to 0