Problem : Find f (t)dt.It follows from the chain rule and the fundamental theorem of calculus that
|f (t)dt = 2f (t)dtf (x)|
Problem : Find all antiderivatives of f (x) = 1/(1 + x) + 2 cos(2x).We guess the antiderivative
|F(x) = log(1 + x) + sin(2x)|
and check that F'(x) = f (x). All other antiderivatives must be of the form F(x) + c for some constant c.
Problem : Compute (3x2 + 7)dx using the fundamental theorem of calculus.We choose x3 + 7x as an antiderivative of 3x2 + 7. The fundamental theorem of calculus then gives
|3x2 + 7dx||=||x3 +7x|-24|
|=||(43 +7(4)) - ((- 2)3 + 7(- 2))|