STANDARD ELEMENTARY INTEGRAL FORMS - General Methods of Integration

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^vdv 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^vdv,

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 = 2x, then dv = 2dx. 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 2a^3x dx.

Solution. Let v = 3x; then dv = 3dx. 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^2x³·x² dx

9. e^x (e^x + 1)dx

10.

11. xk^x² 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 3ax dx.

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

EXAMPLE 3. Find csc²x²·x dx.

Solution. Let v = x²; then dv = 2x 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