Use the fundamental theorem of Calculus to find the derivative of the following function. F (x) = Z √x x et(sin t + t2) dt

Question

F(x) = Z √x x e^t(sin t + t^2) dt

Answer 1

∫^x_√xe^t(sin(t)+t²)dt

From fundamental theorem of calculus part 1:

d/dt∫_h(x)^g(x) f(t)dt =f(g(x))g'(x)-f(h(x))h'(x)

Where:

g(x) = x

g'(x) = 1

h(x) = √x

h'(x) = 1/(2√x)

f(t) = _xe^t(sin(t)+t²)

Just substitute in values into the FTC to finish