The table of integrals and the simplest rules
Answers are for parents and teachers.
1
Compute ∫ (8x³ − 5x + 2/x) dx (x > 0).
2x⁴ − 5x²/2 + 2 ln x + C.
2
Find ∫ (cos 3x − 4 sin(2x + 1)) dx.
(1/3) sin 3x + 2 cos(2x + 1) + C.
3
Find ∫ x²/(x³ + 1) dx (x > −1) by substitution.
u = x³ + 1, x² dx = du/3: (1/3) ln(x³ + 1) + C.
4
Using sin²x = (1 − cos 2x)/2 compute ∫ sin²x dx and check the answer by differentiating.
x/2 − (sin 2x)/4 + C; derivative: 1/2 − cos 2x/2 = sin²x.
5
Find the antiderivative of f(x) = eˣ + 2x whose graph passes through (0, 3).
F = eˣ + x² + C; 1 + C = 3 ⇒ C = 2; F = eˣ + x² + 2.