Calculus 2: Indefinite Integrals and Fundamental Integration Rules

Opening Vincentian Prayer

  • Invocation and Text:

    • Prayer opens in the name of the Father, the Son, and the Holy Spirit. Amen.

    • "Dear Lord, teach me the things that are important: to be generous with your gifts, compassionate to those who have less, just in the face of unfair circumstances, to ruin the world's values when they contradict my principles, and to stand firm when things don't go my way and when they do."

    • "May nothing else matter except faith in your goodness, my neighbors' and mine, hope that things can get better, and charity that always sets things right."

    • "May your special love for the poor—the mark of my uniquely Vincentian education—be the work I excel in, the standard I constantly refer to, and my courage when I meet you someday."

    • "O Mary, conceived without sin, pray for us who have recourse to thee."

    • "Saint Vincent de Paul, pray for us."

    • In the name of the Father, the Son, and the Holy Spirit. Amen.

Instructor Profile and Academic Background

  • Instructor Name: Mister Rafael J. Eusebio Junior

  • Current Academic Roles:

    • Part-time Faculty at Adamson University (teaching since 2019).

    • Part-time Faculty at Universidad de Manila (UDM).

  • Professional Full-Time Role:

    • Acting Department Manager of the Financial Risk Department at Academic Fund.

    • Statistician and Researcher.

  • Educational Attainment:

    • Bachelor of Science in Mathematics with a Minor in Computer Science from Universidad de Manila (UDM), graduated in 2008.

    • Master of Science in Mathematics Education from Polytechnic University of the Philippines (PUP Manila).

    • Currently pursuing a Doctor of Philosophy (PhD) in Mathematics Education at Centro Escolar University (CEU Manila).

  • Certifications and Credentials:

    • Licensed Professional Teacher (LPT).

    • Civil Service Examination Passer.

Diagnostic Assessment: Differential Calculus Review

  • Prerequisite Knowledge Check: Differential calculus serves as the mandatory prerequisite for integral calculus (Calculus 2). A diagnostic evaluation establishes mastery of fundamental differentiation rules.

  • Diagnostic Problems and Step-by-Step Solutions:

    • Problem 1: Find the derivative of f(x)=x3+2x−5f(x) = x^3 + 2x - 5.

      • Solution: Apply the Power Rule ddx[un]=nun−1\frac{d}{dx}\big[u^n\big] = n u^{n-1} and the Constant Rule ddx[c]=0\frac{d}{dx}\big[c\big] = 0

      • f′(x)=3x3−1+2(1)−0f'(x) = 3x^{3-1} + 2(1) - 0

      • f′(x)=3x2+2f'(x) = 3x^2 + 2

    • Problem 2: Find the derivative of f(x)=exf(x) = e^x.

      • Solution: Apply the Exponential Rule for base ee

      • f′(x)=exf'(x) = e^x

    • Problem 3: Find the derivative of f(x)=lˉnˉ(x)f(x) = \bar{l}\bar{n}(x).

      • Solution: Apply the Logarithmic Rule for natural logarithm

      • f′(x)=1xf'(x) = \frac{1}{x}

    • Problem 4: Find the derivative of f(x)=sˉiˉnˉ(x)f(x) = \bar{s}\bar{i}\bar{n}(x).

      • Solution: Apply the Basic Trigonometric Differentiation Rule

      • f′(x)=cˉoˉsˉ(x)f'(x) = \bar{c}\bar{o}\bar{s}(x)

    • Problem 5: Find the derivative of f(x)=cˉoˉsˉ(3x2)f(x) = \bar{c}\bar{o}\bar{s}(3x^2).

      • Solution: Apply the Chain Rule dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

      • Let u=3x2u = 3x^2, so f(u)=cˉoˉsˉ(u)f(u) = \bar{c}\bar{o}\bar{s}(u)

      • ddu[cˉoˉsˉ(u)]=−sˉiˉnˉ(u)=−sˉiˉnˉ(3x2)\frac{d}{du}\big[\bar{c}\bar{o}\bar{s}(u)\big] = -\bar{s}\bar{i}\bar{n}(u) = -\bar{s}\bar{i}\bar{n}(3x^2)

      • dudx=ddx[3x2]=6x\frac{du}{dx} = \frac{d}{dx}\big[3x^2\big] = 6x

      • f′(x)=−sˉiˉnˉ(3x2)×6x=−6xsˉiˉnˉ(3x2)f'(x) = -\bar{s}\bar{i}\bar{n}(3x^2) \times 6x = -6x\bar{s}\bar{i}\bar{n}(3x^2)

    • Problem 6: Find the derivative of f(x)=(2x2−3)(5x2+3)f(x) = (2x^2 - 3)(5x^2 + 3).

      • Solution: Apply the Product Rule ddx[u×v]=u×dvdx+v×dudx\frac{d}{dx}\big[u \times v\big] = u \times \frac{dv}{dx} + v \times \frac{du}{dx}

      • Let u=2x2−3u = 2x^2 - 3 and v=5x2+3v = 5x^2 + 3

      • dudx=4x\frac{du}{dx} = 4x

      • dvdx=10x\frac{dv}{dx} = 10x

      • f′(x)=(2x2−3)(10x)+(5x2+3)(4x)f'(x) = (2x^2 - 3)(10x) + (5x^2 + 3)(4x)

      • f′(x)=(20x3−30x)+(20x3+12x)f'(x) = (20x^3 - 30x) + (20x^3 + 12x)

      • f′(x)=40x3−18xf'(x) = 40x^3 - 18x

    • Problem 7: Find the derivative of f(x)=3x−22x+1f(x) = \frac{3x - 2}{2x + 1}.

      • Solution: Apply the Quotient Rule ddx[uv]=v×dudx−u×dvdxv2\frac{d}{dx}\bigg[\frac{u}{v}\bigg] = \frac{v \times \frac{du}{dx} - u \times \frac{dv}{dx}}{v^2}

      • Let u=3x−2u = 3x - 2 and v=2x+1v = 2x + 1

      • dudx=3\frac{du}{dx} = 3

      • dvdx=2\frac{dv}{dx} = 2

      • f′(x)=(2x+1)(3)−(3x−2)(2)(2x+1)2f'(x) = \frac{(2x + 1)(3) - (3x - 2)(2)}{(2x + 1)^2}

      • f′(x)=(6x+3)−(6x−4)(2x+1)2f'(x) = \frac{(6x + 3) - (6x - 4)}{(2x + 1)^2}

      • f′(x)=6x+3−6x+4(2x+1)2f'(x) = \frac{6x + 3 - 6x + 4}{(2x + 1)^2}

      • f′(x)=7(2x+1)2f'(x) = \frac{7}{(2x + 1)^2}

    • Problem 8: Find the derivative of y=(3x2+1)5y = (3x^2 + 1)^5

      • Solution: Apply the Chain Rule

      • y′=5(3x2+1)5−1×ddx[3x2+1]y' = 5(3x^2 + 1)^{5-1} \times \frac{d}{dx}\big[3x^2 + 1\big]

      • y′=5(3x2+1)4×(6x)y' = 5(3x^2 + 1)^4 \times (6x)

      • y′=30x(3x2+1)4y' = 30x(3x^2 + 1)^4

Course Syllabus and Scope (Calculus 2 - Integral Calculus)

  • Course Details: Calculus 2 / Integral Calculus is a 3-unit academic course.

  • Term-Wide Topic Outline:

    • Weeks 1–4: Indefinite Integrals & Basic Integration Formulas

      • Indefinite integral as antiderivatives: definitions and fundamental properties.

      • Basic Integration Formulas: General Power Formula.

      • Integrals leading to logarithmic functions and exponential functions.

      • Integrals leading to trigonometric functions, inverse trigonometric functions, and hyperbolic functions.

    • Weeks 5–6: Techniques of Integration

      • Integration by Parts technique.

      • Transformation by trigonometric formulas and identities.

    • Weeks 7–8: Integration by Substitution

      • Algebraic substitution techniques.

      • Trigonometric substitution techniques.

    • Weeks 9–10: Integration by Partial Fractions

      • Case 1: Distinct linear factors.

      • Case 2: Repeated linear factors.

      • Case 3: Distinct quadratic factors.

      • Case 4: Repeated quadratic factors.

    • Week 11: Definite Integrals and Improper Integrals

      • Definite integrals: definition, properties, evaluation, and theorems on odd and even functions.

      • Change of limits corresponding to change of variables.

      • Evaluation of improper integrals.

    • Weeks 13–16: Applications of Integration and Multiple Integrals

      • Area of plane regions in rectangular coordinates.

      • Volume of solids of revolution: Circular Disc method, Circular Ring / Washer method, and Cylindrical Shell method.

      • Physical applications: Work, Hydrostatic Pressure, and Hydrostatic Force.

      • Surface trace and multiple integrals as volume: trace planes, sphere, cylinder, and quadric surfaces.

      • Double integrals and triple integrals.

Course Grading System and Administrative Policies

  • Grading System Component Breakdown:

    • Periodic Examinations (Prelim, Midterm, Final Exam): 40%40\%

    • Quizzes (Minimum of 2 quizzes per grading period): 30%30\%

    • Assignments, Seatwork, Research Work, and Projects: 30%30\%

  • Final Semester Grade Calculation:

    • Semester Grade=(0.30×Prelim Grade)+(0.30×Midterm Grade)+(0.40×Final Grade)\text{Semester Grade} = (0.30 \times \text{Prelim Grade}) + (0.30 \times \text{Midterm Grade}) + (0.40 \times \text{Final Grade})

  • Academic and Classroom Policies:

    • Passing Threshold: Official passing grade standard is 70%70\%, though conditional passing curves down to 65%65\%\text{--}68%68\% may apply depending on overall performance distribution.

    • Single Weekly Session Structure: Because the class meets only once per week, time utilization is maximized. Meetings feature lectures immediately followed by quizzes, or quizzes followed by lectures.

    • Assignments and Board Work: Assignments are assigned weekly and solved publicly via student board work during the following class meeting.

    • Recitation / Participation Bonus: Active participation and successful board work add bonus points directly onto quiz scores.

    • Required Course Materials: A dedicated notebook for lecture notes and solutions is required; scientific or advanced calculators are allowed for problem-solving.

    • Examination Format: Comprehensive evaluations containing a combination of True/False, Multiple Choice, and pure Problem-Solving items.

Fundamentals of Anti-Differentiation and Integration

  • Definition: Integration (or anti-differentiation) is the mathematical process of finding a original function F(x)F(x) from its given derivative or differential f(x)f(x).

  • Mathematical Concept:

    • A function FF is called an antiderivative of a function ff on an interval II if F′(x)=f(x)F'(x) = f(x) for all xx in II

    • Demonstration 1: Given f(x)=5x3+x2+3f(x) = 5x^3 + x^2 + 3

      • Differentiating yields f′(x)=15x2+2xf'(x) = 15x^2 + 2x

      • Reversing the derivative through integration restores the variable structure 5x3+x25x^3 + x^2, but the specific constant 33 loses its identity during differentiation. Thus, it is replaced by an arbitrary constant CC: 5x3+x2+C5x^3 + x^2 + C

    • Demonstration 2: Given g(x)=5x3+x2−20g(x) = 5x^3 + x^2 - 20

      • Differentiating yields g′(x)=15x2+2xg'(x) = 15x^2 + 2x

      • Integrating 15x2+2x15x^2 + 2x yields 5x3+x2+C5x^3 + x^2 + C

  • Indefinite Integral Notation:

    • ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C

    • ∫\int: The Integral Sign.

    • f(x)f(x): The Integrand (the function being integrated).

    • dxdx: The Differential indicating that xx is the variable of integration.

    • F(x)F(x): The Particular Integral or Particular Antiderivative.

    • CC: The Arbitrary Constant of Integration.

    • F(x)+CF(x) + C: The General Antiderivative or Indefinite Integral.

  • Examples of Indefinite Integrals:

    • Algebraic: ∫2x dx\int 2x\,dx

    • Trigonometric: ∫cˉoˉsˉ(x) dx\int \bar{c}\bar{o}\bar{s}(x)\,dx

    • Exponential: ∫2e2x dx\int 2e^{2x}\,dx

    • Hyperbolic: ∫3sˉeˉcˉhˉ2(3t) dt\int 3\bar{s}\bar{e}\bar{c}\bar{h}^2(3t)\,dt

Core Properties of the Indefinite Integral

  • Property 1 (Integral of a Differential):

    • ∫d(f(x))=f(x)+C\int d(f(x)) = f(x) + C

    • The integral operator and differential operator cancel each other out.

  • Property 2 (Constant Multiple Rule):

    • ∫kf(x) dx=k∫f(x) dx\int k f(x)\,dx = k \int f(x)\,dx

    • Where kk is any real number constant.

  • Property 3 (Sum and Difference Rule):

    • ∫[f1(x)±f2(x)±⋯±fn(x)] dx=∫f1(x) dx±∫f2(x) dx±⋯±∫fn(x) dx\int \big[f_1(x) \pm f_2(x) \pm \dots \pm f_n(x)\big]\,dx = \int f_1(x)\,dx \pm \int f_2(x)\,dx \pm \dots \pm \int f_n(x)\,dx

  • Basic Application Problems:

    • Example 1: Evaluate ∫dy\int dy

      • Apply Property 1: ∫dy=y+C\int dy = y + C

    • Example 2: Evaluate ∫dθ\int d\theta

      • Apply Property 1: ∫dθ=θ+C\int d\theta = \theta + C

    • Example 3: Evaluate ∫d(x2)\int d(x^2)

      • Apply Property 1: ∫d(x2)=x2+C\int d(x^2) = x^2 + C

    • Example 4: Evaluate ∫3 dx\int 3\,dx

      • Apply Property 2 and Property 1: ∫3 dx=3∫dx=3x+C\int 3\,dx = 3 \int dx = 3x + C

    • Example 5: Evaluate ∫5 dv\int 5\,dv

      • Apply Property 2 and Property 1: ∫5 dv=5∫dv=5v+C\int 5\,dv = 5 \int dv = 5v + C

    • Example 6: Evaluate ∫12 dm\int \frac{1}{2}\,dm

      • Apply Property 2 and Property 1: ∫12 dm=12∫dm=12m+C\int \frac{1}{2}\,dm = \frac{1}{2} \int dm = \frac{1}{2}m + C

    • Example 7: Evaluate ∫−10 dy\int -10\,dy

      • Apply Property 2 and Property 1: ∫−10 dy=−10∫dy=−10y+C\int -10\,dy = -10 \int dy = -10y + C

    • Example 8: Evaluate ∫(x3−4x2+x+1) dx\int (x^3 - 4x^2 + x + 1)\,dx

      • Apply Property 3: ∫x3 dx−∫4x2 dx+∫x dx+∫1 dx\int x^3\,dx - \int 4x^2\,dx + \int x\,dx + \int 1\,dx

The Simple Power Integration Formula

  • Definition & Formula:

    • ∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

    • Restriction: This formula holds true for all real values of nn except n=−1n = -1. If n=−1n = -1, the denominator n+1=0n + 1 = 0, rendering the expression mathematically undefined.

  • Fully Detailed Worked Examples:

    • Example 1: Evaluate ∫x4 dx\int x^4\,dx

      • Apply simple power rule with n=4n = 4:

      • ∫x4 dx=x4+14+1+C\int x^4\,dx = \frac{x^{4+1}}{4+1} + C

      • ∫x4 dx=x55+C=15x5+C\int x^4\,dx = \frac{x^5}{5} + C = \frac{1}{5}x^5 + C

    • Example 2: Evaluate ∫dθθ5\int \frac{d\theta}{\theta^5}

      • Rewrite using laws of exponents: 1θ5=θ−5\frac{1}{\theta^5} = \theta^{-5}

      • ∫θ−5 dθ=θ−5+1−5+1+C\int \theta^{-5}\,d\theta = \frac{\theta^{-5+1}}{-5+1} + C

      • ∫θ−5 dθ=θ−4−4+C\int \theta^{-5}\,d\theta = \frac{\theta^{-4}}{-4} + C

      • Express with positive exponent: −14θ−4+C=−14θ4+C-\frac{1}{4}\theta^{-4} + C = -\frac{1}{4\theta^4} + C

    • Example 3: Evaluate ∫x23 dx\int \sqrt[3]{x^2}\,dx

      • Convert radical expression to fractional exponent: x23=x2/3\sqrt[3]{x^2} = x^{2/3}

      • ∫x2/3 dx=x2/3+123+1+C\int x^{2/3}\,dx = \frac{x^{2/3 + 1}}{\frac{2}{3} + 1} + C

      • Convert integer 11 to common denominator fraction 33\frac{3}{3}:

      • 23+33=53\frac{2}{3} + \frac{3}{3} = \frac{5}{3}

      • ∫x2/3 dx=x5/353+C\int x^{2/3}\,dx = \frac{x^{5/3}}{\frac{5}{3}} + C

      • Apply division of fractions rule ABC=A×CB\frac{A}{\frac{B}{C}} = A \times \frac{C}{B}:

      • ∫x2/3 dx=35x5/3+C\int x^{2/3}\,dx = \frac{3}{5}x^{5/3} + C

    • Example 4: Evaluate ∫(y−4−3y−5) dy\int (y^{-4} - 3y^{-5})\,dy

      • Separate terms using Property 3: ∫y−4 dy−3∫y−5 dy\int y^{-4}\,dy - 3\int y^{-5}\,dy

      • Integrate first term: y−4+1−4+1=y−3−3=−13y3\frac{y^{-4+1}}{-4+1} = \frac{y^{-3}}{-3} = -\frac{1}{3y^3}

      • Integrate second term: −3(y−5+1−5+1)=−3(y−4−4)=3−4y−4  ⟹  +34y4-3 \left(\frac{y^{-5+1}}{-5+1}\right) = -3 \left(\frac{y^{-4}}{-4}\right) = \frac{3}{-4 y^{-4}} \implies +\frac{3}{4y^4}

      • Combine results and add arbitrary constant: ∫(y−4−3y−5) dy=−13y3+34y4+C\int (y^{-4} - 3y^{-5})\,dy = -\frac{1}{3y^3} + \frac{3}{4y^4} + C

    • Example 5: Evaluate ∫2t−t3t dt\int \frac{2t - t^3}{t}\,dt

      • Simplify algebraic fraction prior to integrating:

      • 2t−t3t=2tt−t3t=2−t2\frac{2t - t^3}{t} = \frac{2t}{t} - \frac{t^3}{t} = 2 - t^2

      • ∫(2−t2) dt=∫2 dt−∫t2 dt\int (2 - t^2)\,dt = \int 2\,dt - \int t^2\,dt

      • Apply Property 2 and Simple Power Rule:

      • ∫2 dt−∫t2 dt=2t−t2+12+1+C\int 2\,dt - \int t^2\,dt = 2t - \frac{t^{2+1}}{2+1} + C

      • ∫2t−t3t dt=2t−t33+C\int \frac{2t - t^3}{t}\,dt = 2t - \frac{t^3}{3} + C

    • Example 6: Evaluate ∫(1+3x)(2−x) dx\int (1 + 3x)(2 - x)\,dx

      • Expand binomial products using the FOIL method:

      • (1+3x)(2−x)=1(2)+1(−x)+3x(2)+3x(−x)=2−x+6x−3x2=2+5x−3x2(1 + 3x)(2 - x) = 1(2) + 1(-x) + 3x(2) + 3x(-x) = 2 - x + 6x - 3x^2 = 2 + 5x - 3x^2

      • Set up term-by-term integrals: ∫2 dx+5∫x dx−3∫x2 dx\int 2\,dx + 5\int x\,dx - 3\int x^2\,dx

      • Integrate term 1: 2x2x

      • Integrate term 2: 5(x1+11+1)=52x25 \left(\frac{x^{1+1}}{1+1}\right) = \frac{5}{2}x^2

      • Integrate term 3: −3(x2+12+1)=−3(x33)=−x3-3 \left(\frac{x^{2+1}}{2+1}\right) = -3 \left(\frac{x^3}{3}\right) = -x^3

      • Combine terms: ∫(1+3x)(2−x) dx=2x+52x2−x3+C\int (1 + 3x)(2 - x)\,dx = 2x + \frac{5}{2}x^2 - x^3 + C

    • Example 7: Evaluate ∫(v2−a2)3v4 dv\int (v^2 - a^2)^3 v^4\,dv

      • Expand the binomial cube (v2−a2)3(v^2 - a^2)^3 using algebraic binomial expansion:

        1. Cube the first term: (v2)3=v6(v^2)^3 = v^6

        2. Thrice the product of the square of the first term and the second term: 3(v2)2(−a2)=−3a2v43(v^2)^2(-a^2) = -3a^2 v^4

        3. Thrice the product of the first term and the square of the second term: 3(v2)(−a2)2=3a4v23(v^2)(-a^2)^2 = 3a^4 v^2

        4. Cube the last term: (−a2)3=−a6(-a^2)^3 = -a^6

        5. Expanded expression: (v6−3a2v4+3a4v2−a6)(v^6 - 3a^2 v^4 + 3a^4 v^2 - a^6)

      • Multiply expanded binomial by v4v^4:

      • (v6−3a2v4+3a4v2−a6)v4=v10−3a2v8+3a4v6−a6v4(v^6 - 3a^2 v^4 + 3a^4 v^2 - a^6)v^4 = v^{10} - 3a^2 v^8 + 3a^4 v^6 - a^6 v^4

      • Integrate term-by-term with respect to variable vv (treating aa as constant):

      • ∫v10 dv=v1111\int v^{10}\,dv = \frac{v^{11}}{11}

      • −3a2∫v8 dv=−3a2(v99)=−a2v93-3a^2 \int v^8\,dv = -3a^2 \left(\frac{v^9}{9}\right) = -\frac{a^2 v^9}{3}

      • 3a4∫v6 dv=3a4(v77)=3a4v773a^4 \int v^6\,dv = 3a^4 \left(\frac{v^7}{7}\right) = \frac{3a^4 v^7}{7}

      • −a6∫v4 dv=−a6(v55)=−a6v55-a^6 \int v^4\,dv = -a^6 \left(\frac{v^5}{5}\right) = -\frac{a^6 v^5}{5}

      • Final integrated expression:

      • ∫(v2−a2)3v4 dv=v1111−a2v93+3a4v77−a6v55+C\int (v^2 - a^2)^3 v^4\,dv = \frac{v^{11}}{11} - \frac{a^2 v^9}{3} + \frac{3a^4 v^7}{7} - \frac{a^6 v^5}{5} + C

Student Questions and Academic Discussion

  • Question on Diagnostic Problem Scenarios:

    • Query: Request for clarity on steps for solving Diagnostic Question 6 (f(x)=(2x2−3)(5x2+3)f(x) = (2x^2 - 3)(5x^2 + 3)).

    • Response: Can be expanded first into 10x4−9x2−910x^4 - 9x^2 - 9 and then differentiated directly into 40x3−18x40x^3 - 18x, or evaluated directly through the Product Rule uv′+vu′u v' + v u'

  • Question on Course Materials & Notes:

    • Query: Is there a required format or device for maintaining class notes?

    • Response: A physical notebook is required, though individual strategy for taking and organizing personal study notes is at the student's discretion.

  • Question on Exam Format & Solving:

    • Query: Will examinations consist purely of solving problems?

    • Response: Examinations combine True or False, Multiple Choice, and pure Problem Solving.

  • Question on Calculator Usage:

    • Query: Are calculators permitted for advanced problem-solving?

    • Response: Advanced scientific calculators are permitted and encouraged for solving complex integral calculations.

  • Question on Course Grade Adjustment (Nurturance Curve):

    • Query: Is there a nurturance or grade adjustments similar to Calculus 1?

    • Response: Nurturance and curve adjustments depend on overall situational class performance. Final grades around 65%65\%\text{--}68%68\% may be curved up to passing standard (70%70\%).

  • Class Communication Setup:

    • A class Facebook Messenger Group Chat (GC) is created for official updates. The instructor's personal Facebook account is named Rafael Eusebio.