MTH1170 Integration by Parts

Preliminary The method of Integration by Parts is the integral version of The Product Rule for differentiation. We use it to “undo” The Product Rule and can be considered the Anti-Product Rule. The goal of Integration by Parts is to make a difficult integral easier to solve by substituting the variables u & v for parts of the integration.

How does it work? We start by writing out The Product Rule:

How does it work? Next we integrate both sides of the equation with respect to x and simplify:

How does it work? We rearrange the terms:

How does it work? This equation says that integrals with the same form as the left side of this equation, will have solutions with the same form as the right side of the equation. Notice that the solution has two parts. This is the reason for the name of this technique. 

How does it work? This equation has the functions f, g- prime, f-prime, and g. To keep track of everything, and to make the integration easier we will make a substitution:

How does it work? We can now make the substitution arriving at the formula for Integration by Parts

How does it work? To solve integrals using this method we need to pick a u, and a dv, to substitute. Find the derivatives of u and the integral of dv, and then substitute this into the formula for Integration by Parts.

How to pick u and dv? Pick a ‘u’ so that the integral of dv is possible.  Pick a ‘u' such that the derivative of u is less complex. The derivative of u should be “nicer” than u.  Pick a ‘dv’ such that the integral of dv does not get more complex. The integral of dv should be at minimum no more complex than dv.

LIATE A good way to decide on which part of the function should become ‘u’ is to us the mnemonic LIATE: L - Logs I - Inverse A - Algebraic T - Trig E – Exponential The first of these functions to appear in the integrand should become u.

The Procedure Pick the function within the integral to be substituted for u. Everything that remains within the integral (including the differential element dx) will be defined as dv. Differentiate u and solve for du. Integrate dv and solve for v. Following the formula for Integration by Parts, write out the solution to the original integral in terms of u and v. Substitute u Substitute v Substitute du Solve the integral that remains.

Example Solve the following integral:

Example Continued:

Example Solve the following integral:

Example Continued:

Example Solve the following integral:

Example Continued:

Example Solve the following integral:

Example Continued:

Example Solve the following integral:

Example Continued:

Example Continued: We need to use Integration by Parts again to solve the remaining integral.

Example Continued:

Example Continued Now that we have solved the remaining integral we can plug it back into the previous equation.:

Example Solve the following integral:

Example Continued:

Example Continued: We need to use Integration by Parts again to solve the remaining integral.

Example Continued:

Example Continued: Now that we have solved the remaining integral we can plug it back into the previous equation.