Problem : Find f (t)dt .
It follows from the chain rule and the fundamental theorem of calculus that
f (t)dt = 2 f (t)dt f (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) |
Problem : Compute (3x ^{2} + 7)dx using the fundamental theorem of calculus.
We choose x ^{3} + 7x as an antiderivative of 3x ^{2} + 7 . The fundamental theorem of calculus then gives
3x ^{2} + 7dx | = | x ^{3} +7x|_{-2} ^{4} | |
= | (4^{3} +7(4)) - ((- 2)^{3} + 7(- 2)) | ||
= | 114 |