ITLR - Integration Techniques and L'Hopital's Rule Module Overview

Integration Techniques and L'Hopital's Rule Module Overview

Introduction

Since integration is not as straightforward as differentiation, no rules exist absolutely guaranteeing that an indefinite integral of a function can be found. However, several integration techniques such as integration by parts are available for obtaining indefinite integrals of some complicated functions. Yet another curious situation confronts you when the evaluation of indeterminate limits is needed. L'Hopital's Rule is invaluable for determining limits involving indeterminate forms.

Essential Questions

In what situation would integration by parts be helpful in finding an antiderivative?
What characterizes an indeterminate form?
Why is L'Hopital's Rule an important technique for finding limits?

Key Terms

The following key terms will help you understand the content in this module.

Indeterminate form - A product, quotient, difference, or power of functions that are undefined when the argument of the function has a certain value, because one or both of the functions are zero or infinite. The indeterminate forms include $LaTeX: 0^0,0/0,1^{\infty},\infty-\infty,\infty/ \infty ,0.\infty , \text{&}\infty ^{0}$ $0^0,0/0,1^{\infty},\infty-\infty,\infty/ \infty ,0.\infty , \text{&}\infty ^{0}$

Integration by parts - A technique for finding integrals involving a product of two functions. The integration by parts formula expresses one integral in terms of a second integral $LaTeX: \int f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int g\left(x\right)f'\left(x\right)dx$ $\int f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int g\left(x\right)f'\left(x\right)dx$

L'Hopital's rule - A technique for evaluating indeterminate forms of the type 0/0 or $LaTeX: \pm \infty /\pm \infty$ $\pm \infty /\pm \infty$ . The rule says that the limit of a quotient of differentiable functions is equal to the limit of the quotient of their derivatives, provided the derivative of the denominator does not equal 0.

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