Calculus II: Course Overview and Integration by Parts

Course Logistics and Grading Policies

• Grade components include online homework at 9% and module quizzes at 2% of the overall grade.

• Built-in drop policy: 4 lowest homework assignments, 4 lowest module quizzes, and 2 lowest discussion quizzes are dropped. Due to the class size of 1,300 students, no makeups are granted.

• Grading scale cutoffs are determined at the end of the semester based on class performance; an overall average of 92.5 guarantees an A in the worst-case scenario.

• Office hours are held on Mondays and Fridays immediately following class. Students may attend office hours for any TA listed on Canvas under Course Orientation.

• Online homework is completed through WebAssign. In-person help for WebAssign registration and technical issues is available Thursday and Friday of week 1, and Thursday of week 2, along with a 1-800 support number.

• Drop-in tutoring is available at the Math Learning Center located in Van Vleck, 2nd Basement (B2), operating from 9 to 5.

• Canvas course tabs:

  • Discussion Materials: Contains PDFs of discussion worksheets, worksheet solutions, and quiz solutions.

  • Piazza: Online forum for course structure and content questions (e.g., section 7.3 problem 2, exam questions from fall eighteen). Posts are anonymous to peers but visible to instructors.

  • Direct Email: Used strictly for personal or private matters, such as McBurney accommodations, family schedule conflicts, or serious illness.

Course Schedule and Exams

• Weekly Schedule:

  • Monday, Wednesday, Friday: Lecture with instructor.

  • Tuesday, Thursday: Discussion section with TA.

  • Thursday: Homework assignments are due; discussion quizzes take place on scheduled Thursdays.

  • Sunday: Module quizzes are due (initial module quizzes listed for the 6th on the schedule have a WebAssign due date of the 13th).

• Exam Schedule:

  • Midterm exams occur on Tuesday nights with built-in makeups for valid conflicts (e.g., overlapping chemistry lab).

  • Final exam takes place on Saturday night (not Saturday morning at 07:25AM).

• Canvas Announcements: Students should enable email notifications for Canvas announcements (posted 1 to 2 times per week).

Questions & Discussion

• Exam Accommodations:

  • Question: Are exams taken at the exact same time when using extra time accommodations?

  • Response: Exams occur on the same day and around the same time, but within a longer extended time window.

• WebAssign Access:

  • Question: Where is the link to register for WebAssign?

  • Response: Under the second toggle on Canvas labeled "Textbook and WebAssign Information."

• Student Introductions:

  • Students introduced themselves and discussed majors (data science, CS 200, CS 400, electrical engineering), dorms (Pronsedge, Lakeshore, Bradley, Chabula), and hometowns (New Berlin near Milwaukee/UW Milwaukee).

Integration by Parts

• Prerequisite Review:

  • Basic integrals: $\int \cos(x)\,dx = \sin(x) + C$, $\int \frac{1}{x}\,dx = \ln(|x|) + C$.

  • $u$-substitution is used when two multiplied functions inside an integral are directly related by a derivative.

• Definition & Formula:

  • Integration by parts is used to integrate the product of two functions $f(x)$ and $g'(x)$ that do not share a derivative relationship:   $\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx$

  • Using $u$ and $v$ notation ($u = f(x)$, $v = g(x)$, $du = f'(x)\,dx$, $dv = g'(x)\,dx$):   $\int u\,dv = u v - \int v\,du$

  • Definite Integral Formula:   $\int_{a}^{b} u\,dv = [u v]_{a}^{b} - \int_{a}^{b} v\,du$

• Function Selection Priority (LILATE Acronym for Choosing $u$):

  1. L - Logarithmic functions: $\ln(x)$, $\log_2(x)$

  2. I - Inverse trigonometric functions: $\sin^{-1}(x)$, $\cos^{-1}(x)$, $\tan^{-1}(x)$ (or $\arcsin(x)$, $\arccos(x)$, $\arctan(x)$)

  3. A - Algebraic functions: $x$, $x^2$, $\sqrt{x} = x^{\frac{1}{2}}$, $x^{\frac{2}{3}}$

  4. T - Trigonometric functions: $\sin(x)$, $\cos(x)$, $\tan(x)$, $\csc(x)$, $\sec(x)$, $\cot(x)$

  5. E - Exponential functions: $e^x$, $5^x$

  • Whatever component remains in the integrand (including $dx$) forms $dv$.

Worked Examples

• Example 1: Evaluate $\int x \cos(x)\,dx$

  • Let $u = x$ $\rightarrow$ $du = dx$

  • Let $dv = \cos(x)\,dx$ $\rightarrow$ $v = \sin(x)$

  • Apply Integration by Parts:   $\int x \cos(x)\,dx = x \sin(x) - \int \sin(x)\,dx$   $= x \sin(x) - (-\cos(x)) + C$   $= x \sin(x) + \cos(x) + C$

• Example 2: Evaluate $\int x e^x\,dx$

  • Let $u = x$ $\rightarrow$ $du = dx$

  • Let $dv = e^x\,dx$ $\rightarrow$ $v = e^x$

  • Apply Integration by Parts:   $\int x e^x\,dx = x e^x - \int e^x\,dx$   $= x e^x - e^x + C$