The Natural Logarithmic Function: Integration

Theorem 5.5 Log Rule for Integration
Let u be a differentiable function of x.

Because du = u’dx, the second formula can also be written as

Ex 1 Find the indefinite integral.

Ex 2 Find the indefinite integral.

Ex 3 Find the indefinite integral.

Ex 4 Find the indefinite integral by u-substitution. (Hint: Let u be the
denominator of the integrand .)

Ex 5 Find the indefinite integral. You may have to use long division first .

Guide lines for Integration

1. Learn a basic list of integ ration formulas . (Including those given in
this section you now have 12 formulas: the Power Rule , the Log Rule,
and ten trigonometric rules . By the end of Section 5.7, this list will
have expanded to 20 basic rules.)

2. Find an integration formula that resembles all or part of the integrand,
and, by trial and error, find a choice of u that will make the integrand
conform to the formula.

3. If you cannot find a u- substitution that works , try altering the
integrand. You might try a trigonometric identity, multiplication and
division
by the same quantity, or addition and subtraction of the same
quantity. Be creative.

4. If you have access to computer software that will find antiderivatives
symbolically , use it.

Ex 6 Find the indefinite integral:

Integrals of the Six Basic Trigonometric Functions

Ex 7 Solve the differential equation. Use a graphing utility to graph three
solutions , one of which passes through the given point. ,(0,4)

Ex 8Evaluate the definite integral. Use a graphing utility to verify your
result.

Ex 9 Find the area of the region bounded by the graphs of the equations .
Use a graphing utility to verify your result.

Ex 10 Find the average value of the function over the given interval.

5.2 Homework # 1 – 73 e.o.o. and 85, 87, 91

 Prev Next

Start solving your Algebra Problems in next 5 minutes!

 Algebra Helper Download (and optional CD) Only \$39.99 Click to Buy Now: OR
2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of March 24th you will pay only \$39.99 instead of our regular price of \$74.99 -- this is \$35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1\$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

• simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
• factoring and expanding expressions
• finding LCM and GCF
• (simplifying, rationalizing complex denominators...)
• solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
• solving a system of two and three linear equations (including Cramer's rule)
• graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
• graphing general functions
• operations with functions (composition, inverse, range, domain...)
• simplifying logarithms
• basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
• arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

 Algebra Helper Download (and optional CD) Only \$39.99 Click to Buy Now: OR
2Checkout.com is an authorized reseller
of goods provided by Sofmath

 "It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..." C. Sievert, KY

 Sofmath 19179 Blanco #105-234San Antonio, TX 78258 Phone: (512) 788-5675Fax: (512) 519-1805
 Home   : :   Features   : :   Demo   : :   FAQ   : :   Order Copyright © 2004-2018, Algebra-Answer.Com.  All rights reserved.