Integration by parts applies to both definite and indefinite integrals To integrate products involving sin ( a x ) , sin ( b x ) , cos ( a x ) , and cos ( b x ) , use the substitutions

The integration-by-parts formula allows the exchange of one integral for another, possibly easier, integral Integrals of trigonometric functions can be evaluated by the use of various strategies.

integration by parts — a technique of integration that allows the exchange of one integral for another using the formula ∫ ​ u d v = u v - ∫ ​ v d u. improper integral — an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The…. midpoint rule — a rule that uses a Riemann sum of the form M n = ∑ i = 1 n f ( m i ) Δ x , where m i is the midpoint of the i th subinterval to approximate ∫ a b f ( x ) d x. Simpson’s rule — a rule that approximates ∫ a b f ( x ) d x using the integrals of a piecewise quadratic function. The approximation S n to ∫ a b f ( x ) d x is given by S n = Δ x 3 ( f ( x 0 ) +….