Integration Techniques: Trigonometric Substitution and Partial Fractions

Trigonometric Substitution in Integration

Overview of Trigonometric Substitution

Trigonometric substitution is a powerful technique for evaluating integrals involving square roots of quadratic expressions. By substituting trigonometric functions for variables, certain integrals can be simplified and solved more easily.

• Key Idea: Replace algebraic expressions with trigonometric identities to simplify the integrand.

• Common Substitutions:

  • For $\sqrt{a^2 - x^2}$, use $x = a \sin \theta$

  • For $\sqrt{a^2 + x^2}$, use $x = a \tan \theta$

  • For $\sqrt{x^2 - a^2}$, use $x = a \sec \theta$

• Reference Triangle: Drawing a reference triangle helps convert back to $x$ after integration.

Reference Triangles for Trig Substitution

Reference triangles are used to relate trigonometric functions back to the original variable after integration. The sides of the triangle are chosen based on the substitution used.

Substitution	Triangle Sides	Expression	Reference Triangle Built From
$x = a \sin \theta$	Opposite: $x$, Hypotenuse: $a$	$\sqrt{a^2 - x^2}$	$\sin \theta = \frac{x}{a}$
$x = a \tan \theta$	Opposite: $x$, Adjacent: $a$	$\sqrt{a^2 + x^2}$	$\tan \theta = \frac{x}{a}$
$x = a \sec \theta$	Hypotenuse: $x$, Adjacent: $a$	$\sqrt{x^2 - a^2}$	$\sec \theta = \frac{x}{a}$

Important Note: The trig integrals learned in section 8.3 are extremely important for these problems.

Partial Fractions in Integration

Introduction to Partial Fractions

Partial fraction decomposition is a method for integrating rational functions, where the degree of the numerator is less than the degree of the denominator. The denominator is factored, and the integrand is expressed as a sum of simpler fractions.

• Linear Factors: For $\int \frac{P(x)}{Q(x)} dx$, where $Q(x)$ factors into distinct linear terms:

  • $\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

• Linear Factors with Multiplicity Greater than 1:

  • $\frac{1}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$

• Quadratic Factors That Cannot Be Reduced:

  • $\frac{1}{(x^2 + bx + c)} = \frac{Ax + B}{x^2 + bx + c}$

• Solving for Constants: The coefficients $A$, $B$, $C$, etc., are found by equating coefficients or substituting values for $x$.

• Helpful for: Integrating rational functions where direct integration is not possible.

Helpful Integral Rules and Trigonometric Identities

Some integral formulas and trigonometric identities are especially useful for trigonometric substitution and partial fractions.

• $\int \tan u \, du = -\ln|\cos u|$

• $\int \sec u \, du = \ln|\sec u + \tan u|$

• $\int \cot u \, du = \ln|\sin u|$

• $\int \csc u \, du = -\ln|\csc u + \cot u|$

Remember to use these rules when integrating functions after substitution.

Trigonometric Identities Useful for Integration

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

These identities can help simplify many integrals, not just those involving trigonometric substitution.

Example: Using Trigonometric Substitution

Suppose you need to evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$. Use the substitution $x = a \sin \theta$, so $dx = a \cos \theta d\theta$ and $\sqrt{a^2 - x^2} = a \cos \theta$. The integral becomes:

$\int \frac{a \cos \theta d\theta}{a \cos \theta} = \int d\theta = \theta + C$

Convert back to $x$ using $\theta = \arcsin \left( \frac{x}{a} \right)$.

Example: Partial Fraction Decomposition

Integrate $\int \frac{1}{x^2 - 1} dx$. Factor the denominator: $x^2 - 1 = (x-1)(x+1)$. Write:

$\frac{1}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1}$

Solve for $A$ and $B$:

$1 = A(x+1) + B(x-1)$

Set $x=1$: $1 = A(2) + B(0) \Rightarrow A = \frac{1}{2}$ Set $x=-1$: $1 = A(0) + B(-2) \Rightarrow B = -\frac{1}{2}$

So:

$\int \frac{1}{x^2 - 1} dx = \frac{1}{2} \int \frac{1}{x-1} dx - \frac{1}{2} \int \frac{1}{x+1} dx = \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C$

Additional info: The notes emphasize the importance of knowing trigonometric identities and integral rules for successful application of these techniques. These methods are foundational for solving a wide variety of integrals encountered in Calculus II and beyond.