May 19, 2026 - Trigonometric Substitution and Integration Techniques Overview (Concise)

Review of Trigonometric Identities and Unit Circle

• Pythagorean Identities: The fundamental identity is $\sin^2(x) + \cos^2(x) = 1$. By dividing this by $\cos^2(x)$, we derive $\tan^2(x) + 1 = \sec^2(x)$. Dividing the original by $\sin^2(x)$ yields $1 + \cot^2(x) = \csc^2(x)$.
• Unit Circle (Quadrant I):
  • $0$: $\cos(0) = 1$, $\sin(0) = 0$
  • $\pi/6$: $\cos(\pi/6) = \sqrt{3}/2$, $\sin(\pi/6) = 1/2$
  • $\pi/4$: $\cos(\pi/4) = \sqrt{2}/2$, $\sin(\pi/4) = \sqrt{2}/2$
  • $\pi/3$: $\cos(\pi/3) = 1/2$, $\sin(\pi/3) = \sqrt{3}/2$
  • $\pi/2$: $\cos(\pi/2) = 0$, $\sin(\pi/2) = 1$

Trigonometric Substitution Strategies

• Substitution Choices: The choice of substitution depends on the radical form present in the integral:
  • For $\sqrt{a^2 + x^2}$, use $x = a \tan(\theta)$.
  • For $\sqrt{a^2 - x^2}$, use $x = a \sin(\theta)$.
  • For $\sqrt{x^2 - a^2}$, use $x = a \sec(\theta)$.
• Algebraic Refinement: Sometimes an expression like $\sqrt{1 + 4x^2}$ requires rewriting as $\sqrt{1 + (2x)^2}$ to identify the substitution $2x = \tan(\theta)$.
• Simplification: After substitution, reference triangles ($\text{SOH CAH TOA}$) are necessary to convert the solution back into the original variable $x$.

Integration of Secant Powers

• Power Reduction Formula: To integrate $\int \sec^n(\theta) \,d\theta$, use the formula:     $\int \sec^n(\theta) \,d\theta = \frac{1}{n-1} \sec^{n-2}(\theta) \tan(\theta) + \frac{n-2}{n-1} \int \sec^{n-2}(\theta) \,d\theta$
• Strategy: When dealing with a polynomial of secants, reduce the largest power first. This often allows the result to be combined with other terms in the integral before applying further reductions.

Rational Functions and Long Division

• Applicability: Polynomial long division is used when the degree of the numerator is higher than or equal to the degree of the denominator.
• Process: Divide the numerator by the denominator to obtain a quotient and a remainder. Re-express the integral as:     $\int (\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}) \,dx$
• Example Outcome: The integral of a term like $\frac{3}{x+1}$ results in a natural log: $3 \ln|x+1| + C$.

Partial Fraction Decomposition

• Setup: For a rational function with a composite denominator, factor the denominator and set up a sum of simpler fractions: $\frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+1}$.
• Solving for Coefficients:
  • Heaviside Method: Plug in the roots of the denominator to solve for individual variables ($A, B, C$) quickly.
  • System of Equations: Alternatively, expand the numerator and compare coefficients of like terms (e.g., equating all $x^2$ terms to zero if no $x^2$ exists in the target numerator).
• Integration: Partial fractions typically resolve into natural logarithms or $\arctan$ functions. For example, $\int \frac{1}{x^2+25} \,dx$ leads to an arc-tangent integral.

Questions & Discussion

• Question (Dropped Term): A student noted that a power reduced from $\sec^7$ to $\sec^5$ should have changed a sign when moved in the calculation.
• Response: The instructor acknowledged a "dropped negative" and corrected values such as adding coefficients to get $17/6$.
• Question (Method Priority): A student asked if the test will specify which method to use (e.g., trig sub vs. other techniques).
• Response: The instructor stated the test will primarily require students to set up the problem and will likely not include extremely long calculations (like many power reductions) but may specifically ask for integration by parts or trigonometric substitution to verify proficiency in those methods.