Chapter 9 Calculus Techniques for the Elementary Functions

9.2 Integration by parts – A way to integrate Products Integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals.

Integration by parts – A way to integrate Products If where u and v are differentiable functions of x then: Formula for integration by parts.

Integration by parts – A way to integrate Products Suppose we want to find the volume of the solid formed by rotating the region under around the y-axis. From aside: Aside Let

Integration by parts – A way to integrate Products Examples: Let

9.3 Rapid Repeated Integration by Parts Example: UdV Reason for the sign change. mult integrate

UdV

 Using Trig properties to make original reappear. + -

 Reassociation between Steps reassociate Integrating with substitution method

9.4 Reduction Formulas  Objective is to find a generic formula for

9 – 6 Integration by Trig Substitution Reminder: 1. Since this reminds us of Pythagoras, let’s label a triangle and attempt to change into integration of trig. Key to success is deciding whether the variable x is the leg or the hypotenuse. If we choose x as a leg we can choose which leg. Vertical or horizontal will affect which trig function we will use later on. x

9 – 6 Integration by Trig Substitution 2. When changing from must also change the bounds.

9 – 7 Integration of Rational Functions by Partial Fractions We are wanting to ‘undo’ the common denominator Let x=1 so that we get rid of B and just have A left

SHORTCUT!! Simply place finger over (x-1) when letting x=1 and solve to get A. Then place finger over (x-10) when solving for B.

Integration of Rational Functions by Partial Fractions Examples: 1. This is an example of a rational function in proper form. That is lower degree than. Improper form With improper form we need to perform long division first. Partial fractions will work only with proper form.

Integration of Rational Functions by Partial Fractions One problem that we may encounter is that when we factor the denominator, we find some factors repeated, that is occurring with multiplicities greater than 1. Repeated linear factors 4. Trick to partial fractions:

Integration of Rational Functions by Partial Fractions Unfactorable Quadratics: 5. Trick:

9 – 8 Integrals of the Inverse Trig Functions “Like climbing mount Everest, they are interesting more from the standpoint that they can be done, rather than because they are of great practical use.” In 4 – 5 we found derivatives of inverse trig by implicit differentiation.

9 – 8 Integrals of the Inverse Trig Functions Integrate using parts. + - Aside: our calc and the back of the book use the notion sgnx Treat this as a constant even though we do not know the sign. HW 1 – 9 odd

9 – 10 Improper Integrals Improper Integrals: Upper or lower limit of integration is infinite. Integrand is discontinuous for at least one value of x at or between the limits of integration. An improper integral converges to a certain number if each applicable limit shown below is finite. Otherwise, the integral diverges. Is discontinuous at x=c in

Improper Integrals ** If the integrand seems to approach zero as x gets very large or small, then the integral might converge. Conversely, if the integrand grows without bound as grows without bound, then the integral definitely diverges. Examples: 1.For the improper integral a. Graph the integrand, and tell whether or not the integral might converge. b. If the integral might converge, find out whether or not it really does, and if so, to what limit it converges. Graph looks to be approaching 0 and therefore might converge.

Improper Integrals 2. For the improper integral a. Graph the integrand, and tell whether or not the integral might converge. b. If the integral might converge, find out whether or not it really does, and if so, to what limit it converges. 3. For the improper integral a. Graph the integrand, and tell whether or not the integral might converge. b. If the integral might converge, find out whether or not it really does, and if so, to what limit it converges.

9.11 Miscellaneous Integrals and Derivatives Differentiation  Sum  Product  Quotient  Composite  Implicit  Power Function  Exponential Function  Logarithmic function

Differentiation Con’t  Logarithmic  Trigonometric Function  Inverse Trig Function – differentiate Implicitly

Integration  Sum  Product  Reciprocal Function  Power Function  Power of a Function

Integration Continued  Square root of quadratic: Trig Sub  Rational Algebraic Expression: Convert to sum by long division, partial fractions  Inverse Function: Integrate by parts