### 7.5: Integration of Transcendental Functions

A transcendental number is a number that is not the root of any integer polynomial. A transcendental function, similarly, is a function that cannot be written using roots and the arithmetic found in polynomials. We address exponential, logarithmic, and hyperbolic functions here, having covered the integration and differentiation of trigonometric functions previously.

### 7.5.1: Exponential Functions

Read Section 8.3 (pages 441 through 447). This chapter recaps the definition of the number e and the exponential function and its behavior under differentiation and integration.

Work through problems 1-12. When you are done, check your solutions against the answers provided.

### 7.5.2: Natural Logarithmic Functions

Read Section 8.5 (pages 454 through 459). This chapter reintroduces the natural logarithm (the logarithm with base e) and discusses its derivative and antiderivative. Recall that you can use these properties of the natural log to extrapolate the same properties for logarithms with arbitrary bases by using the change of base formula.

Watch "Video 31: The Natural Logarithmic Function" through the 5th slide (marked 5 of 8). Then watch "Video 32: The Exponential Function."

The first short video gives one definition of the natural logarithm and derives all the properties of the natural log from that definition. It shows examples of limits, curve sketching, differentiation, and integration using the natural log. We will return to this video later to watch the last three slides. The second video explains the number e, the exponential function and its derivative and antiderivative, curve sketching using the exponential function, and how to perform similar operations on power functions with other bases using the change of base formula.

Click "Index" button for Book II. Scroll down to "3. Transcendental Functions," and click button 137 (Logarithm, Definite Integrals). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

### 7.5.3: Hyperbolic Functions

Watch these videos. The creator of the video pronounces "sinh" as "chingk." The more usual pronunciation is "sinch."

Read Section 8.4 (pages 449 through 453). In this chapter, you will learn the definitions of the hyperbolic trig functions and how to differentiate and integrate them. The chapter also introduces the concept of capital accumulation.

Work through each of the sixteen examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.