Calculus II Study Guide: Integration Techniques and Hyperbolic Functions (Ch. 5.1–8.9)

Integration and the Fundamental Theorem of Calculus

Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus connects differentiation and integration, providing a way to evaluate definite integrals and understand antiderivatives.

• FTC Part 1: If $F(x) = \int_a^x f(t)\,dt$, then $\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$.

• FTC Part 2: If $F(x)$ is any antiderivative of $f(x)$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

• Key Skills:

  • Recognize when to flip the bounds of an integral (which introduces a negative sign).

  • Split integrals at points of discontinuity or to simplify computation.

  • Apply the chain rule when the upper limit is a function of $x$.

• Example: $\frac{d}{dx} \int_1^{x^2} \sin t\,dt = \sin(x^2) \cdot 2x$ (by the chain rule).

Integration Techniques

u-Substitution

u-Substitution is a method for finding antiderivatives by substituting part of the integrand with a new variable $u$.

• Process:

  1. Let $u = g(x)$, where $g(x)$ is a function inside the integrand.

  2. Compute $du = g'(x)dx$.

  3. Rewrite the integral in terms of $u$ and $du$.

  4. Integrate and substitute back for $x$.

• Special Antiderivatives:

  • $\int \tan x\,dx = -\ln|\cos x| + C$

  • $\int \sec x\,dx = \ln|\sec x + \tan x| + C$

• Polynomial Long Division: Use when the degree of the numerator is greater than or equal to the denominator before integrating rational functions.

Hyperbolic Functions

Definitions and Properties

Hyperbolic functions are analogs of trigonometric functions but are based on exponential functions.

• Definitions:

  • $\sinh x = \frac{e^x - e^{-x}}{2}$

  • $\cosh x = \frac{e^x + e^{-x}}{2}$

  • $\tanh x = \frac{\sinh x}{\cosh x}$

• Derivatives:

  • $\frac{d}{dx} \sinh x = \cosh x$

  • $\frac{d}{dx} \cosh x = \sinh x$

  • $\frac{d}{dx} \tanh x = 1 - \tanh^2 x = \text{sech}^2 x$

• Antiderivatives:

  • $\int \sinh x\,dx = \cosh x + C$

  • $\int \cosh x\,dx = \sinh x + C$

  • $\int \tanh x\,dx = \ln|\cosh x| + C$

• Applications: Use definitions to prove identities and compute derivatives/antiderivatives.

• Example: To find $\frac{d}{dx} \tanh x$, use the quotient rule on $\frac{\sinh x}{\cosh x}$.

Advanced Integration Techniques

Long Division of Polynomials

Before integrating a rational function, perform long division if the numerator's degree is greater than or equal to the denominator's.

• Process: Divide the numerator by the denominator to write the integrand as a sum of a polynomial and a proper fraction.

• Example: $\int \frac{x^3 + 2x^2 + 1}{x^2 + 1}\,dx$

Integration by Parts

Integration by parts is based on the product rule for differentiation and is used to integrate products of functions.

• Formula: $\int u\,dv = uv - \int v\,du$

• Choosing $u$ and $dv$: Use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to select $u$.

• Special Cases:

  • Integrals involving $\ln x$ or $\arctan x$ often require integration by parts.

  • "Bootstrapping": When repeated application of integration by parts cycles back to the original integral, solve algebraically.

• Example: $\int x e^x\,dx$

Trigonometric Integrals

Integrals involving powers of trigonometric functions often require the use of identities to simplify.

• Pythagorean Identities:

  • $\sin^2 x + \cos^2 x = 1$

  • $\tan^2 x + 1 = \sec^2 x$

• Half-Angle Identities:

  • $\sin^2 x = \frac{1}{2}(1 - \cos 2x)$

  • $\cos^2 x = \frac{1}{2}(1 + \cos 2x)$

• Double Angle Identity: $\sin(2x) = 2\sin x \cos x$

• Application: Use these identities to reduce the power of trigonometric functions before integrating.

Trigonometric Substitution

Trigonometric substitution is used to integrate expressions involving square roots of quadratic expressions.

• When to Use:

  • $\sqrt{a^2 - x^2}$: Substitute $x = a\sin \theta$

  • $\sqrt{a^2 + x^2}$: Substitute $x = a\tan \theta$

  • $\sqrt{x^2 - a^2}$: Substitute $x = a\sec \theta$

• Important: Always replace $dx$ with the correct expression in terms of $d\theta$.

• Back-Substitution: Use a right triangle to convert $\theta$ back to $x$ after integrating.

Partial Fraction Decomposition (PFD)

Partial fraction decomposition is used to integrate rational functions by expressing them as a sum of simpler fractions.

• Process:

  1. Perform long division if the numerator's degree is greater than or equal to the denominator's.

  2. Factor the denominator completely.

  3. Set up partial fractions for each factor:

     • Linear factors: $\frac{A}{x - r}$

     • Repeated factors: $\frac{A_1}{x - r} + \frac{A_2}{(x - r)^2} + \cdots$

     • Irreducible quadratics: $\frac{Ax + B}{x^2 + bx + c}$

  4. Solve for coefficients and integrate each term.

• Example: $\int \frac{1}{x^2 - 1}\,dx = \int \left(\frac{1}{2(x-1)} - \frac{1}{2(x+1)}\right)dx$

Mixed Integration Practice

Section 8.6 emphasizes the importance of practicing a variety of integration techniques to identify the appropriate method for each problem.

• Key Point: Be able to recognize which technique (u-substitution, integration by parts, trigonometric substitution, partial fractions, etc.) is suitable for a given integral.

Improper Integrals

Improper integrals involve limits where the interval of integration is infinite or the integrand has a vertical asymptote.

• Definition: $\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

• Vertical Asymptotes: If the integrand is undefined at a point in the interval, split the integral and use limits.

• Convergence/Divergence: Determine if the limit exists (converges) or not (diverges).

• Common Functions with Asymptotes: Rational functions, $\tan x$, $\ln x$.

• L'Hospital's Rule: May be needed to evaluate limits arising in improper integrals.

• Example: $\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b \to \infty} \left(-\frac{1}{x}\right)\Big|_1^b = 1$

Summary Table: Integration Techniques and When to Use Them

Technique	When to Use	Key Steps
u-Substitution	Integrand contains a function and its derivative	Let $u = g(x)$, rewrite in terms of $u$, integrate, substitute back
Integration by Parts	Product of two functions (e.g., $x e^x$, $x \sin x$)	Choose $u$ and $dv$, use $\int u\,dv = uv - \int v\,du$
Trigonometric Substitution	Integrals with $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$	Substitute $x$ with $a\sin \theta$, $a\tan \theta$, or $a\sec \theta$ as appropriate
Partial Fractions	Rational functions with factorable denominators	Decompose into simpler fractions, integrate each term
Long Division	Numerator degree ≥ denominator degree in rational function	Divide first, then integrate the result

Additional info: This guide omits sections 8.7 and 8.8 as per the instructions in the original notes. For best results, students should practice a variety of problems from each section and review the relevant textbook examples.