## The Calculus Primer (2011)

### Part XII. General Methods of Integration

### Chapter 46. STANDARD ELEMENTARY INTEGRAL FORMS

**12—8. Standard Integrals.** Thus far we have discussed five so-called standard elementary integrals, which we restate below for convenience:

* dx* = x + C [1]

* a dv = a dv***[2]**

** ( du + dv − dw) = du + dv − dw**

**[3]**

We now give a list of the remaining standard integral forms for reference:

**12—9. The Forms a^{v}dv and e^{v} dv.** These formulas are seen to be true from the following considerations. We know that

*d*(*a ^{v}*) = log

*a·a*

^{v}·dv;integrating both sides, we get:

*a ^{v}* = log

*a*·

*a*

^{v}·dv,or *a ^{v}* = log

*a a*

^{v}dv;hence

In a similar manner, since

*d*(*e ^{v}*)

*=e*

^{v}dv,it follows that * e ^{v} dv* =

*e*

^{v}+ C.The reader can readily verify these results by differentiation.

EXAMPLE 1. Find *e*^{2x} *dx.*

*Solution.* If we let *v* = 2*x,* then *dv* = 2*dx.* Now, in order to make the expression *e*^{2x} *dx* conform to the standard form *e ^{v} dv,* since

*dv*=

*2dx*when

*v = 2x,*we insert the factor 2 before the

*dx,*and the factor before the integral sign; thus

but *e ^{v} dv* =

*e*+

^{v}*C*, hence

EXAMPLE 2. Find 2*a*^{3x} *dx.*

*Solution.* Let *v* = 3*x*; then *dv* = 3*dx*. Insert 3 before the *dx,* and before the integral sign; then

EXAMPLE 3. Find *e*^{cos x} (sin *x*) *dx.*

*Solution.* Let *v* = cos *x*; *dv* = − sin *x* *dx*; then

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

**EXERCISE 12—4**

*Find the following integrals; check by differentiation:*

**1.** *e*^{4z} *dx*

**2.** *ka ^{mx} dx*

**3.** *e*^{sin x} (*cox x*) *dx*

**4.** *e*^{2 sin x} (cos *x*) *dx*

**5.** *e*^{−3x} *dx*

**6.** *a ^{x/n} dx*

**7.** *a*^{2x−1} *dx*

**8.** *e*^{2x}^{3}·*x*^{2} *dx*

**9.** * e ^{x}* (

*e*1)

^{x}+*dx*

**10.**

**11.** *xk ^{x}*

^{2}

*dx*

**12.** (*e ^{x}*

^{/2}+

*e*

^{−x/2})

*dx*

**12—10. Standard Forms [8] through [13].** These formulas follow immediately from the corresponding formulas for differentiation, as given in §5—9, §5—10, §5—11, §5—12, equations [6]–[11]. They may be verified simply and directly by differentiation.

EXAMPLE 1. Find cos 3*ax dx.*

*Solution.* Let *v* = *3ax;* then *dv* = 3*a dx.*

EXAMPLE 3. Find csc^{2}*x*^{2}·*x* *dx.*

*Solution.* Let *v* = *x*^{2}; then *dv =* 2*x dx*.

**EXERCISE 12—5**

*Find the following integrals; check by differentiation:*

*Verify the following:*

**12—11. Standard Forms [14]–[17].** These four formulas may be proved by transforming the integrand in each case so that we may apply equation [5]; the proofs follow.

*Form* [14]:

But − log cos *v* = − = − log 1 + log sec *v* = log sec *v*;

hence, tan *v dv =* log sec *v.*

*Form* [15]: In a similar manner,

*Form* [16]: To transform the integrand so that it will be in the form we write:

*Form* [17]: This may be derived in a manner similar to the proof for [16]. We leave it as an exercise for the reader; multiply the integrand csc *v* by the fraction .

**12—12. Forms [18]–[23].** The formulas for the standard forms [18]–[23] may be derived by suitable transformations or substitutions. Formulas [22] and [23] follow at once from the corresponding formulas for differentiation.

EXAMPLE 1. Find

*Solution.* Let *v*^{2} = 9*x*^{2}, and *a*^{2} = 4; then *v* = 3*x*, *dv* = 3*dx,* and *a* = 2.

From formula [18], we get:

EXAMPLE 2. Find

*Solution.* Let *v*^{2} = 4*x*^{2}, and *a*^{2} = 25; then *v =* 2*x, dv* = 2*dx,* and *a* = 5.

From formula [19], we get:

EXAMPLE 3. Find

*Solution.* Let *v*^{2} = 4*x*^{2}, and *a*^{2} = 9; then *v* = 2*x, dv* = *d*(2*x*), and *a* = 3.

From formula [20]:

EXAMPLE 4. Find .

*Solution.* Let *v*^{2} = 5*x*^{2}, and *a*^{2} = 3; then *v* = *x*, *dv* = *dx*, and *a* = .

From formula [19]:

EXAMPLE 5. Find

*Solution.* Rewrite as follows:

Here *v* = 2*x, a* = 5, *dv* = 2*dx.*

From formula [22]:

EXAMPLE 6. Find

*Solution.* Let *v*^{2} = *x*^{4}, and *a*^{2} = 36; then *v* = *x*^{2}, *dv* = 2*x* *dx*, and *a* = 6.

From formula [20]:

EXAMPLE 7. Find

*Solution.* Rewrite as follows, completing the square in the denominator:

Now apply formula [18], where *v* = *x +* 2, *a =* 4.

**EXERCISE 12—6**

*Find the following integrals; check by differentiation:*

*Verify the following:*

**EXERCISE 12—7**

**Review**

*Find the following integrals:*

**13.** e^{cos x.}sin *x dx*

**14.** The slope of a certain curve is given by = 6*x*^{2} − 10*x* + 8. If the curve passes through the point (2,10), find its equation.

**15.** If = 12*x* + 6, find *y* in terms of *x* if it is known that when *x* = 2, = 28, and when *x* = −3, *y* = − 1.