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 non-negative, 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
To find the derivative of a definite integral using the Fundamental Theorem of Calculus, you can follow these steps:
Step 1: Write the integral in a proper form.
The definite integral from x^3 to sqrt(x) of (sqrt(t)) * sin(t) dt can be written as ∫(x^3 to sqrt(x)) sqrt(t) * sin(t) dt.
Step 2: Find the antiderivative.
To find the antiderivative of sqrt(t) * sin(t), you can use integration by parts. Let's split the integrand into two parts: u = sqrt(t) and dv = sin(t) dt.
Differentiating u gives: du/dt = (1/2)t^(-1/2).
Integrating dv gives: v = -cos(t).
Now, you can apply the formula for integration by parts: ∫ u dv = uv - ∫ v du.
The integral becomes: ∫ sqrt(t) * sin(t) dt = -sqrt(t) * cos(t) - ∫ (-cos(t) * (1/2)t^(-1/2)) dt.
Step 3: Simplify the integral.
By simplifying, the integral becomes: -sqrt(t) * cos(t) + (1/2) ∫ (t^(-1/2)) cos(t) dt.
Step 4: Evaluate the definite integral.
Now, you need to evaluate the definite integral from x^3 to sqrt(x). To do this, substitute the limits of integration into the simplified integral:
((-sqrt(sqrt(x)) * cos(sqrt(x)) + (1/2) ∫ (t)^(-1/2) cos(t) dt) - (-sqrt(x^3) * cos(x^3) + (1/2) ∫ (t)^(-1/2) cos(t) dt)).
Notice that the terms with the indefinite integral cancel out, and you are left with:
- sqrt(sqrt(x)) * cos(sqrt(x)) + sqrt(x^3) * cos(x^3).
That's the derivative of the definite integral from x^3 to sqrt(x) of (sqrt(t)) * sin(t) dt, obtained using the Fundamental Theorem of Calculus.