Integration by parts comes from the chain rule for derivatives. Integration by parts always evaluates an integral directly. It helps if u' is simpler than u and v is simpler than v'. Integration by parts does not apply to definite integrals. It may be necessary to apply integration by parts more than once.
Most functions have elementary antiderivatives.
Partial fractions requires the decomposition of a quotient intotwo or more fractions and comes from the solution of several equations inseveral unknowns. The midpoint rule is the average of the left and right sums. The trapezoid rule gives an over estimate if the integral is concave up. The midpoint rule gives an overestimate if the integral is concave up. The trapezoid in Figure 7.6 has top edge drawn as a secant line.