## Calculus AB and Calculus BC

## CHAPTER 5 Antidifferentiation

### D. INTEGRATION BY PARTS

**Parts Formula**

The Parts Formula stems from the equation for the derivative of a product:

or, or more conveniently *d*(*uv*) = *u dv* + *v du*.

Hence, *u dv* = *d*(*uv*) − *v du* and integrating gives us or

the Parts Formula. Success in using this important technique depends on being able to separate a given integral into parts *u* and *dv* so that (a) *dv* can be integrated, and (b) *du* is no more difficult to calculate than the original integral.

**EXAMPLE 43**

Find

**SOLUTION:** We let *u* = *x* and *dv* = cos *x dx*. Then *du* = *dx* and *v* = sin *x*. Thus, the Parts Formula yields

**EXAMPLE 44**

Find

**SOLUTION:** We let *u* = *x*^{2} and *dv* = *e*^{x}*dx*. Then *du* = 2*x dx* and *v* = *e** ^{x}*, so We use the Parts Formula again, this time letting

*u*=

*x*and

*dv*=

*e*

^{x}*dx*so that

*du*=

*dx*and

*v*=

*e*

*. Thus,*

^{x}**EXAMPLE 45**

Find *I* =

**SOLUTION:** To integrate, we can let *u* = *e** ^{x}* and

*dv*= cos

*x dx*; then

*du*=

*e*

^{x}*dx*,

*v*= sin

*x*. Thus,

To evaluate the integral on the right, again we let *u* = *e** ^{x}*,

*dv*= sin

*x dx*, so that

*du*=

*e*

^{x}*dx*and

*v*= − cos

*x*. Then,

**EXAMPLE 46**

Find

**SOLUTION:** We let *u* = ln *x* and *dv* = *x*^{4} *dx*. Then, and Thus,

THE TIC-TAC-TOE METHOD. ^{1}

This method of integrating is extremely useful when repeated integration by parts is necessary. To integrate we construct a table as follows:

Here the column at the left contains the successive derivatives of *u*(*x*). The column at the right contains the successive antiderivatives of *v*(*x*) (always with *C* = 0); that is, *v*_{1} (*x*) is the antiderivative of *v*(*x*), *v*_{2} (*x*) is the antiderivative of *v*_{1} (*x*), and so on. The diagonal arrows join the pairs of factors whose products form the successive terms of the desired integral; above each arrow is the sign of that term. By the tic-tac-toe method,

**EXAMPLE 47**

To integrate cos *x dx* by the tic-tac-toe method, we let *u*(*x*) = *x*^{4} and *v*(*x*) = cos *x*, and get the following table:

The method yields

With the ordinary method we would have had to apply the Parts Formula four times to perform this integration.

^{1} This method was described by K. W. Folley in Vol. 54 (1947) of the *American Mathematical Monthly* and was referred to in the movie *Stand and Deliver.*