Integrals and differential equations

Integration runs differentiation backwards. Where a derivative tells you the rate, an integral adds up the change to give you the total. This topic covers the two integration skills that appear on almost every AP Calculus AB exam. The first page is antiderivatives and u-substitution. Every indefinite integral needs a + C, because many functions share the same derivative. With u-substitution, the common slip is changing the variable but keeping the old x bounds. Either convert the bounds to u right away, or switch back to x before you plug anything in. The second page is separable differential equations. You gather the y terms with dy and the x terms with dx, integrate both sides and add one constant. Then comes the step that is easy to forget: use the initial condition to find that constant. Only then have you actually solved the problem. Check each answer by differentiating it and seeing if you get back where you started.