Integration by parts
Integration by parts is a process used to find the integral of a product of functions by using a formula to turn the integral into one that is simpler to compute. Given two functions f(x) and g(x), the formula for integration by parts is as follows:
This formula is somewhat cumbersome, so we can simplify it by making some substitutions. Let u = f(x), v = g(x), du = f'(x)dx, and dv = g'(x)dx. Then, the above formula can be rewritten as:
To use the integration by parts formula, first identify (or select) u and dv and find du and v. Then, plug everything into the formula.
The selection of u and dv is an important part of the process because based on the functions involved in the product, the calculation of one integral may be easier than the other. Also, depending on the selection, the formula may result in an integral that cannot be evaluated.
Selecting u and dv
There is no guaranteed way to determine the ideal selection of u and dv. However, generally, it is best to select a u that becomes simpler when differentiated, and a v that does not become more complicated when integrated. A helpful rule of thumb for achieving this is to use the acronym ILATE; each letter indicates a type of function:
The above functions are generally listed in order of decreasing complexity when differentiated. Thus, given the integral of a product of functions, identify the types of functions involved, and select u as the type of function that appears highest on the list. The other function is dv.
Below are a few examples of using integration by parts.
Use integration by parts to find the following.
x is an algebraic function and sin(x) is a trigonometric function. Algebraic functions are higher in the list than trigonometric functions, so let x = u and sin(x) = dv. Then, v = -cos(x), du = dx, and the integral can be evaluated as follows:
x is an algebraic function and ex is an exponential function. Algebraic functions are higher in the list than exponential functions, so let x = u and ex = dv. Then, v = ex, du = dx, and the integral can be evaluated as follows:
x is an algebraic function and ln(x) is a logarithmic function. Logarithmic functions appear higher in the list than algebraic functions, so let ln(x) = u and x dx = dv. Then, v = , du = , and the integral can be evaluated as follows:
It is worth noting that there are integrals where using integration by parts multiple times is necessary; it is also possible that computing an integral requires the use of integration by parts along with other integration methods.
Let and . Then, and . Substituting these values into the formula for integration by parts results in the following:
The integral in the above result can also be integrated by parts. Let dv = sin(x)dx and u = x. Then, v = -cos(x) and du = dx. Plugging this into the formula:
Plugging this back into the first result: