How to find values of a function based off of it's definite integral (am I thinking about this wrong, or do I just plug in the number?)

Question

F(x)= ∫0 to x sin(2t) dt

 

evaluate F(pi)

For this one I thought I would just plug in sin(2pi) but I got that wrong. It asks for an exact number so I thought I shouldn't give the decimal answer. Is there something I overlooked?

 

and for what interval is F(x) non- negative?

for this one I found the derivative and based on that said from [0, infinity) it would be positive but I got it wrong.

Answer 1

F(x) = ∫₀^x sin(2t)dt

 

Evaluate the definite integral as if x was just a number.

 

F(x) = [-cos(2t) / 2] |₀^x = (1/2)[-cos(2x) + cos(0)] = [1 - cos(2x)]/2

 

F(π) = [1 - cos(2π)]/2 = 0

Answer 2

F(x) = ∫_{(from 0 to x)} sin(2t)dt

 

F(π) = ∫_{(from 0 to π)} sin(2t)dt

 

       = [-½ cos(2t)]_{(from 0 to π)}

 

       = -½[cos(2π) - cos0] = -½[1-1] = 0