THE INTEGRAL

5.7 Further Transcendental Functions

In Section 5.3, we used FTC I to show

We obtain a formula for ln x as a definite integral by setting a = 1 and b = x:

Thus, ln x is equal to an area under the hyperbola y = 1/t (Figure 1).

Figure 5.48

In a similar fashion, we can express sin⁻¹ x as a definite integral using the derivative formula from Section 3.9 (Figure 2):

Figure 5.49

Since sin⁻¹ 0 = 0, we have

On the other hand, the derivative formulas from Section 3.8 yield integration formulas that are useful for evaluating new types of integrals.

Inverse Trigonometric Functions

It is possible (and mathematically, it is more efficient) to take Eq. (1) as the definition of ln x and to define e^x as the corresponding inverse function (see Exercises 78–79).

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In this list, we omit the integral formulas corresponding to the derivatives of cos⁻¹ x, cot⁻¹ x, and csc⁻¹ x because the integrals differ only by a minus sign from those already on the list. For example,

EXAMPLE 1

Evaluate .

Solution This integral is the area of the region in Figure 3. By Eq. (3),

Figure 5.50: The shaded region has an area equal to

EXAMPLE 2 Using Substitution

Evaluate

Solution Notice that can be written as , so it makes sense to try the substitution u = 2x, du = 2 dx. Then

The new limits of integration are . By Eq. (4),

In substitution, we usually define u as a function of x. Sometimes, it is more convenient to define x as a function of u. We do this here, where we set x = 2u.

EXAMPLE 3 Using Substitution

Evaluate .

Solution Let us first rewrite the integrand:

Thus it makes sense to use the substitution . Then and

The new limits of integration are u (0) = 0 and :

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Integrals Involving [em]f[/em]([em]x[/em]) = [em]b[/em][sup][em]x[/em][/sup]

The exponential function f(x) = e^x is particularly convenient because e^x is both its own derivative and its own antiderivative. For other bases b, we have