➗Calculus II Unit 3 – Techniques of Integration

Key Concepts

• Integration is the process of finding the antiderivative of a function
• Antiderivatives are functions whose derivative is the original function
  • Antiderivatives are not unique and differ by a constant $C$
• Definite integrals have fixed limits of integration and represent the area under a curve between those limits
• Indefinite integrals have no fixed limits and represent a family of antiderivatives
• The Fundamental Theorem of Calculus connects differentiation and integration
  • It states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at the limits $a$ and $b$
• Integration techniques are used to evaluate integrals that cannot be solved using basic antiderivative formulas

Integration Strategies

• Recognize the structure of the integrand to determine the appropriate integration technique
• Simplify the integrand by factoring, expanding, or rearranging terms to make integration easier
• Break down complex integrals into simpler parts using properties of integration, such as linearity and additivity
• Use substitution to transform the integrand into a form that can be integrated using known antiderivative formulas
• Apply integration by parts when the integrand is a product of functions, one of which is easier to integrate than the other
• Employ trigonometric substitution for integrals containing $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$
• Decompose rational functions using partial fractions before integrating
• Identify and evaluate improper integrals, which have infinite limits or discontinuities in the integrand

Substitution Method

• The substitution method is used to simplify integrals by introducing a new variable
• Choose a substitution that transforms the integrand into a form that can be integrated using known antiderivative formulas
• Common substitutions include:
  • $u = f(x)$ for integrals of the form $\int f'(x) \cdot g(f(x)) dx$
  • Trigonometric substitutions for integrals containing $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$
• After substituting, rewrite the integral in terms of the new variable $u$
• Integrate the new expression with respect to $u$
• Substitute back the original variable to obtain the antiderivative in terms of $x$

Integration by Parts

• Integration by parts is used to integrate products of functions
• The formula for integration by parts is $\int u dv = uv - \int v du$
• Choose $u$ and $dv$ such that $u$ is easier to differentiate and $dv$ is easier to integrate
  • The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) can help in selecting $u$
• Differentiate $u$ to find $du$ and integrate $dv$ to find $v$
• Substitute $u$, $v$, $du$, and $dv$ into the integration by parts formula
• Evaluate the resulting integral, which may require additional applications of integration by parts or other techniques

Trigonometric Integrals

• Trigonometric integrals involve functions of sine and cosine
• Use trigonometric identities to simplify the integrand:
  • $\sin^2 x + \cos^2 x = 1$
  • $\cos^2 x = \frac{1 + \cos 2x}{2}$
  • $\sin^2 x = \frac{1 - \cos 2x}{2}$
• For integrals of the form $\int \sin^m x \cos^n x dx$:
  • If $m$ is odd, use the substitution $u = \cos x$
  • If $n$ is odd, use the substitution $u = \sin x$
  • If both $m$ and $n$ are even, use the half-angle formulas to reduce the powers
• For integrals of the form $\int \tan^m x \sec^n x dx$, use the substitution $u = \sec x$

Partial Fractions

• Partial fraction decomposition is used to integrate rational functions (quotients of polynomials)
• Decompose the rational function into a sum of simpler fractions
• The decomposition depends on the factors of the denominator:
  • For distinct linear factors, use $\frac{A}{ax + b}$
  • For repeated linear factors, use $\frac{A_1}{(ax + b)^1} + \frac{A_2}{(ax + b)^2} + \cdots$
  • For distinct quadratic factors, use $\frac{Ax + B}{ax^2 + bx + c}$
• Solve for the unknown coefficients ($A$, $B$, etc.) by equating the decomposed form with the original rational function
• Integrate each resulting fraction using appropriate techniques

Improper Integrals

• Improper integrals are integrals with infinite limits or discontinuities in the integrand
• For integrals with infinite limits:
  • Evaluate the limit of the definite integral as the upper or lower limit approaches infinity
  • If both limits are infinite, split the integral at a convenient point and evaluate each part separately
• For integrals with discontinuities:
  • Identify the point(s) of discontinuity
  • Split the integral at the discontinuity and evaluate each part separately
  • If the discontinuity is at an endpoint, evaluate the limit of the definite integral as the endpoint approaches the discontinuity
• Improper integrals may converge (finite value) or diverge (infinite value or undefined)

Applications and Practice

• Area between curves: $\int_a^b [f(x) - g(x)] dx$
• Volume of solids of revolution:
  • Disk method: $V = \int_a^b \pi [f(x)]^2 dx$
  • Shell method: $V = \int_a^b 2\pi x f(x) dx$
• Arc length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$
• Work: $W = \int_a^b F(x) dx$
• Solve a variety of practice problems to reinforce understanding of integration techniques
• Identify the most appropriate technique for each problem based on the structure of the integrand
• Combine multiple techniques when necessary to evaluate complex integrals