CHAPTER 6: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING SECTION 6.3: ANTIDIFFERENTIATION BY PARTS AP CALCULUS AB

What you’ll learn about Product Rule in Integral Form Solving for the Unknown Integral Tabular Integration Inverse Trigonometric and Logarithmic Functions … and why The Product Rule relates to derivatives as the technique of parts relates to antiderivatives.

Section 6.3 – Antidifferentiation by Parts Remember the Product Rule for Derivatives: By reversing this derivative formula, we obtain the integral formula

Section 6.3 – Antidifferentiation by Parts Integration by Parts Formula

Section 6.3 – Antidifferentiation by Parts Example:

Example Using Integration by Parts

Section 6.3 – Antidifferentiation by Parts Try to pick for your u something that will get smaller, or more simple when you take the derivative. Sometimes, you may have to integrate by parts more than one time to get to something you can integrate. Sometimes, integration by parts does not work.

Section 6.3 – Antidifferentiation by Parts Repeated use

Example Repeated Use of Integration by Parts

Example Antidifferentiating ln x

Section 6.3 – Antidifferentiation by Parts Solving for the Unknown Integral Sometimes, when performing this process, we seem to get a circular argument going. When this occurs, collect your like terms, and you can solve for the unknown integral.

Example:

Example Solving for the Unknown Integral

Section 6.3 – Antidifferentiation by Parts Tabular Integration When many repetitions of integration by parts is needed, there is a way to organize the calculations that saves a great deal of work and confusion. It is called tabular integration.

Section 6.3 – Antidifferentiation by Parts Example: (+) (-) (+) (-)