Calculus 2 Integration Techniques and Review

Basic Integration and Integration by Parts

• Example: Integration involving Exponentials and Roots     * Evaluate the integral: $\int \sqrt{x} e^{\sqrt{x}} \, dx$

• Example: Integration by Parts with Polynomial and Trigonometric Factors     * Evaluate the integral: $\int (2x^2 + 6x) \sin(2x) \, dx$

• Example: Integration by Parts with Logarithmic Functions     * Evaluate the integral: $\int x^2 (\ln(x))^2 \, dx$

• Example: Inverse Trigonometric Functions     * Evaluate the integral: $\int \arctan(x) \, dx$

• Example: Looping Integration by Parts     * Evaluate the integral: $\int \sin(x) e^x \, dx$

Trigonometric Integrals

• Example: Product of Sine and Cosine with Odd and Even Powers     * Evaluate: $\int \sin(7x) \cos^8(7x) \, dx$

• Example: Squared Sine Functions     * Evaluate: $\int \sin^2(3x) \, dx$

• Example: Products of Even Powers of Sine and Cosine     * Evaluate: $\int \sin^4(3x) \cos^2(3x) \, dx$

• Example: High Odd Powers of Sine     * Evaluate: $\int \sin^5(2x) \, dx$

• Example: Tangent and Secant Integrals     * Integral 1: $\int \tan^3(x) \sec^2(x) \, dx$     * Integral 2: $\int \tan^3(x) \sec^3(x) \, dx$

• Example: Cotangent and Cosecant Integrals     * Evaluate: $\int \cot^2(x) \csc^2(x) \, dx$

Trigonometric Substitution and Fundamental Identities

• Essential Trigonometric Identities     * Relationship between Sine and Cosine: $\cos^2(\theta) = 1 - \sin^2(\theta)$     * Relationship between Secant and Tangent (Identity 1): $\sec^2(\theta) = 1 + \tan^2(\theta)$     * Relationship between Secant and Tangent (Identity 2): $\tan^2(\theta) = \sec^2(\theta) - 1$

• Example Problems using Trigonometric Substitution     * Integral containing $(x^2 - 9)^{3/2}$: $\int \frac{1}{(x^2 - 9)^{3/2}} \, dx$     * Integral containing $\sqrt{x^2 + 1}$: $\int \frac{1}{x \sqrt{x^2 + 1}} \, dx$     * Integral with square of radical: $\int \frac{1}{(1 + x^2)^2} \, dx$     * Integral with root of negative square: $\int x^3 \sqrt{4 - 9x^2} \, dx$     * Integral with root of positive square: $\int \frac{\sqrt{4x^2 - 9}}{x} \, dx$

Partial Fraction Decomposition (PFD)

• Decomposition Setup (No Coefficient Determination Required)     * Case A: $\frac{-3x + 20}{x^3 + 3x^2 - 10x}$     * Case B: $\frac{x - 3}{x(x+1)^3 (x^2 + 5)}$     * Case C: $\frac{x^2 + 2x + 7}{x^4 + 2x^2 + 1}$     * Case D: $\frac{2x + 1}{(x+3)(x^2 + 1)^2}$

• Integrals Involving Partial Fractions     * Integral 1: $\int \frac{-3x + 20}{x^3 + 3x^2 - 10x} \, dx$     * Integral 2: $\int \frac{1}{x^3 + x^2} \, dx$     * Integral 3: $\int \frac{x^3 - 7x^2 + 2x + 8}{x^3 + 4x^2} \, dx$     * Integral 4: $\int \frac{x+1}{(x+3)(x+4)} \, dx$

Improper Integrals and Comparison Theorem

• Definite and Improper Integrals     * Integration involving substitution and trigonometric bounds: $\int_0^{\pi/2} \frac{\cos(x)}{1 + \sin^2(x)} \, dx$     * Infinite upper bound (Type 1): $\int_1^{\infty} \frac{1}{(x+3)^{1.5}} \, dx$     * Discontinuity at lower bound (Type 2): $\int_0^8 \frac{1}{\sqrt[3]{x}} \, dx$

• Convergence Testing via Comparison Theorem     * Check convergence for: $\int_1^{\infty} \frac{1}{3x^2 - 1} \, dx$     * Check convergence for: $\int_1^{\infty} \frac{e^{-x}}{x^4} \, dx$     * Check convergence for: $\int_0^{\pi} \frac{\sin(x)}{\sqrt{x}} \, dx$

• Special Substitutions     * Evaluate the integral: $\int \frac{1}{3 + 2\cos(2x)} \, dx$

To solve the integral $\int \sqrt{x} e^{\sqrt{x}} \, dx$ using the tabular method of integration by parts, we can follow these steps:

1. Set up the table with derivatives of one function and integrals of the other:

   • Choose:

     • Function (Choose u): $u = e^{\sqrt{x}}$ (since it simplifies upon differentiation)

     • Derivative (du): Differentiate until you reach 0:

     • First derivative: $du = \frac{1}{2\sqrt{x}} e^{\sqrt{x}} \, dx$

     • Second derivative: $du = -\frac{1}{4x^{3/2}} e^{\sqrt{x}} \, dx$

     • Continue this until a simple expression is reached.

   • Choose:

     • Function (Choose dv): $dv = \sqrt{x} \, dx$

     • Integral (v): Integrate:

     • First integral: $v = \frac{2}{3} x^{3/2}$

     • Second integral: $v = \frac{1}{12} x^{5/2}$

2. Construct the Tabular Method:

   • This will give a table that looks like this:

   Derivative	Integral
   e^{\sqrt{x}}	\frac{2}{3} x^{3/2}
   \frac{1}{2\sqrt{x}} e^{\sqrt{x}}	\frac{1}{12} x^{5/2}
   (Continue until 0)	(Continue)

3. Sign Alternation: Attach alternating signs (starting with +) to each term:

   • The first product will be added, the second will be subtracted, and so on.

4. Final Expression:

   • Combine these products according to the pattern established:

   • This will yield:
     $\int \sqrt{x} e^{\sqrt{x}} \, dx = e^{\sqrt{x}} \cdot \frac{2}{3} x^{3/2} - \int (\text{next product}) \, dx + C$

   • Continue simplifying as necessary based on remaining integrals until you arrive at a solution.

This method efficiently organizes the integration by parts process, especially for functions that require repeated applications of the technique.