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$\int \frac{1}{e^{x}+1} d x$ (Hints:(a) multiply the numerator and denominator by $e^{-x}.$(b) Write the numerator as $\left(e^{x}+1\right)-e^{x},$ then divide. (c) Explain the differ-ent results.)

(a) $-\ln \left(1+e^{-x}\right)+c_{1}$(b) $x-\ln \left(e^{x}+1\right)+c_{2}$(c) they differ by a constant

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

00:44

Multiply the numerator and…

00:46

01:33

$\left.\int e^{x+\frac{1}{…

01:44

$\int \frac{x e^{x}}{(1+x)…

02:04

$\int\left[\mathrm{dx} /\l…

0:00

evaluate Integral

01:21

Evaluate the definite inte…

How to evaluate this integ…

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