Integration by parts is not the exclusive strategy for products

Choosing the parts to integrate and differentiate

Products and typical strategies

Review the strategies for products and determining the parts to integrate and differentiate: [SHOW MORE]

Product type

Preliminary thoughts

Strategy for this product type

Examples

polynomial function times basic trigonometric or exponential function (such as sin, cos, exp, cosh, sinh, or a polynomial in such expressions)

The polynomial function can be differentiated, and repeated differentiation keeps making it simpler and simpler until it disappears. The sine or cosine function or exponential function, upon integration, does not get more complicated, and so can be integrated repeatedly.

Take the polynomial function as the part to differentiate, and keep using integration by parts repeatedly till the polynomial disappears.

Trig: (see here for worked out examples) Exp: (see here for worked out examples)

inverse trigonometric function or logarithmic function (or composite of such a function with a polynomial function) times polynomial function (the polynomial function could just be the function , which is invisible).

The polynomial function can easily be both differentiated and integrated. The inverse trigonometric or logarithmic function can be differentiated, bringing it into the algebraic domain.

Choose the inverse trigonometric function or logarithmic function as the part to differentiate.

Simple products: (see here for worked out examples) Products involving composites:

trigonometric function times exponential function OR product of trigonometric functions or power of a trigonometric function that can be treated as a product (and other techniques such as integration by u-substitution don't seem to solve the problem completely)

both functions are easy to differentiate and to integrate, with the complexity remaining the same.