## The Calculus Primer (2011)

### Part XIII. Special Methods of Integration

### Chapter 47. INTEGRATION BY PARTS

**13—1. Need for Special Methods.** The standard forms discussed in the preceding chapter enable us to integrate many expressions that arise. However, other types of expressions may be encountered for which these standard forms will not suffice, and so certain special methods must be employed. One of the most common of these is the method of *integration by parts.*

**13—2. Integration by Parts.** Consider *u* and *v,* which represent functions of *x.* We have:

which may also be written in the differential form

or *d*(*uv*) = *v du + u dv,* (1)

where

Now, by transposing, equation (1) may be written

*u dv* = *d*(*uv*) *−* *v du.* (2)

By integrating both sides of (2), we obtain

* u dv = uv − v du.* [1]

It will be seen that we may make use of equation [1] whenever it is possible to find the integral of *v du.* The method of “integrating by parts” thus amounts to this: if we cannot integrate *f*(*x*) *dx* directly, we try to break the expression *f*(*x*) *dx* into *two* factors or “parts,” say *u* and *dv,* such that the integrals of both *dv* and *v du* are readily found. The method of procedure is suggested by the following:

*f*(*x*) *dx* = *u dv*; * u dv* = *uv −* *v du.* [2]

Frankly, no general rule can be given to indicate the best way of selecting the factors *u* and *dv* in all cases; each example must be studied individually. However, with practice the reader will doubtless develop facility in using this method, which is one of the most useful of all special methods of integration. It may help to remember that we try to select *u,* and the corresponding *dv,* in such a way that not only can we find *v* from *dv,* but also that * v du* is easier to evaluate than the original integral, *f*(*x*) *dx.* In some cases it may be necessary to repeat the process one or more times. The following examples will illustrate the application of the method of integration by parts.

EXAMPLE 1. Find * x* sin *x dx.*

*Solution.* Let *u* = *x,* and *dv =* sin *x dx*; then *du* = *dx,* and *v = * sin *x dx = −* cos *x.*

Substituting in equation [2] above:

*x* sin *x dx* = *−* *x* cos *x −* *−* cos *x dx*

= − *x* cos *x +* sin *x + C.*

EXAMPLE 2. Find *x* log *x dx.*

*Solution.* Let *u* = log *x,* and *dv* = *x dx;* then , and

Substituting in [2]:

EXAMPLE 3. Find *xe ^{x} dx.*

*Solution.* Let *u* = *x,* and *dv* = *e ^{x}*

*dx*; then

*du*=

*dx*, and

*v*= e

*.*

^{x}Hence: * xe ^{x} dx = xe^{x} − e^{x} dx*

*= xe ^{x}* −

*e*(

^{x}+ C = e^{x}*x*−

*1*)

*+ C.*

EXAMPLE 4. Find arc sin *x dx*.

*Solution.* Let *u* = arc sin *x,* and *dv* = *dx;* then and *v* = *x.*

Hence, arc sin *x dx* = *x* arc sin *x* −

But, by §12—6, Example 2, we see that

hence arc sin *x dx* = *x* arc sin

EXAMPLE 5. Find *x*^{2} sin *x dx.*

*Solution.* Let *u* = *x*^{2}*,* and *dv* = sin *x dx*; then *du* = 2*x* *dx*, and *v*= sin *x dx* = − cos *x*.

Hence: *x*^{2} sin *x dx = −x*^{2} *cos x + 2 x cos x dx*. (1)

But, to find the integral *x* cos *x* *dx* in equation (1), we must apply the method of integration by parts again; thus

let *u = x,* and *dv* = cos *x dx;*

*du* = *dx,* and *v* = sin *x*.

Hence, *x* cos *x dx = x* sin *x* — sin *x dx*

= *x* sin *x* + cos *x* + *C*. (2)

Therefore, substituting (2) in (1):

* x*^{2} sin *x dx* = −*x*^{2} cos *x* + 2(*x* sin *x* + 2 cos *x*) + *C*

= −*x*^{2} cos *x* + 2*x* sin *x* + 2 cos *x* + *C′.*

EXAMPLE 6. Find *x*^{2} cos 2*x dx.*

*Solution.* Let *u = x*^{2}, and *dv* = cos 2*x dx;* then *du* = 2*x dx*, and

But, to find the integral *x* sin 2*x* *dx*, we use the method once more:

let *u* = *x*, and *dv* = sin 2*x dx*;

Therefore, substituting (2) in (1):

**EXERCISE 13—1**

*Find the following integrals, using the method of integration by parts:*

**1.** *x* cos *x dx*

**2.** log *x dx*

**3.** *xe*^{2z} *dx*

**4.** * x*^{2}*e ^{x}dx*

**5.** *e ^{x}* cos

*x dx*

**6.** log^{2} *x dx*

**7.** * x*^{2} cos *x dx*

**8.** *x*^{3} log *x dx*

**9.** * θ* sec^{2} *θ dθ*

**10.** arc tan *x dx*

**11.** * x*^{2} *e ^{−x} dx*

**12.** *x*^{2} log *x dx*

**13.** * θ* tan^{2} *θ dθ*

**14.** *z* sec^{2} *z* *dz*

**15.** * e ^{x}* cos 2

*x dx*

**16.** cos *x* log sin *x dx*

**17.** *x*^{2} sin 2*x* *dx*

**18.** * x* cos 2*x dx*