8.1 Indefinite Integration Techniques

Indefinite integration techniques are essential tools for solving complex integrals. These methods, including substitution, integration by parts, partial fractions, and trigonometric strategies, help simplify and break down challenging integrals into manageable parts.

By mastering these techniques, you'll be able to tackle a wide range of integration problems. Each method has its strengths, and knowing when to apply them is key to becoming proficient in calculus and mathematical problem-solving.

Indefinite Integration Techniques

Substitution for indefinite integrals

Top images from around the web for Substitution for indefinite integrals

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Systems by Substitution View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

• 

  CS 360: Lecture 6: Recurrence View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Systems by Substitution View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

1 of 3

Top images from around the web for Substitution for indefinite integrals

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Systems by Substitution View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

• 

  CS 360: Lecture 6: Recurrence View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Systems by Substitution View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

1 of 3

• Introduces a new variable (often $u$) as a function of the original variable $x$ to simplify complex integrals
  • Substitution $u = g(x)$ chosen to simplify the integrand
  • Differential $du$ expressed in terms of $[dx](https://www.fiveableKeyTerm:dx)$ using the chain rule: $du = g'(x)dx$
• Steps to apply substitution:
  1. Identify a suitable substitution $u = g(x)$ that simplifies the integrand
  2. Express the integrand in terms of $u$ and $du$
  3. Integrate the simplified expression with respect to $u$
  4. Substitute back the original variable $x$ to obtain the final result
• Examples of integrals solvable using substitution:
  • $\int x\sqrt{x^2 + 1}dx$ (let $u = x^2 + 1$)
  • $\int e^{2x}\cos(e^x)dx$ (let $u = e^x$)

Integration by parts technique

• Used to integrate products of functions based on the product rule for derivatives
  • Formula for integration by parts: $\int u\,dv = uv - \int v\,du$
• Steps to apply integration by parts:
  1. Identify functions $u$ and $dv$ in the integrand
     • $u$ chosen as a function that becomes simpler when differentiated
     • $dv$ chosen such that it can be easily integrated
  2. Compute $du$ by differentiating $u$ and $v$ by integrating $dv$
  3. Apply the integration by parts formula: $\int u\,dv = uv - \int v\,du$
  4. Repeat the process if necessary until the integral becomes simpler to evaluate
• Examples of integrals solvable using integration by parts:
  • $\int x\sin(x)dx$ (let $u = x$ and $dv = \sin(x)dx$)
  • $\int \ln(x)dx$ (let $u = \ln(x)$ and $dv = dx$)

Partial fractions in integration

• Used to integrate rational functions (quotients of polynomials) by expressing them as a sum of simpler fractions
• Steps to apply partial fraction decomposition:
  1. Factor the denominator of the rational function into irreducible factors
  2. Set up a decomposition with unknown coefficients for each factor in the denominator
     • Linear factors $(x - a)$ use the form $\frac{A}{x - a}$
     • Repeated linear factors $(x - a)^n$ use the form $\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n}$
     • Irreducible quadratic factors $(ax^2 + bx + c)$ use the form $\frac{Bx + C}{ax^2 + bx + c}$
  3. Solve for the unknown coefficients by equating the decomposition to the original rational function and comparing coefficients or evaluating at specific points
  4. Integrate each resulting fraction separately using known integration techniques
• Examples of rational functions integrable using partial fraction decomposition:
  • $\int \frac{2x + 1}{x^2 - 4}dx$
  • $\int \frac{3x - 2}{(x - 1)(x + 2)^2}dx$

Strategies for trigonometric integrals

• Techniques for integrating functions containing sine, cosine, tangent, or their reciprocals:
  • Basic trigonometric integrals:
    • $\int \sin(x)dx = -\cos(x) + C$
    • $\int \cos(x)dx = \sin(x) + C$
    • $\int \sec^2(x)dx = \tan(x) + C$
    • $\int \csc^2(x)dx = -\cot(x) + C$
  • Trigonometric substitution used when the integrand contains $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$
    • Substitute $x = a\sin(\theta)$, $x = a\tan(\theta)$, or $x = a\sec(\theta)$, respectively
    • Simplify the integrand and integrate with respect to $\theta$
    • Substitute back the original variable $x$
  • Integration by parts applied when the integrand is a product of trigonometric and algebraic functions (example: $\int x\cos(x)dx$)
  • Trigonometric identities used to simplify the integrand before integrating (examples: $\sin^2(x) = \frac{1 - \cos(2x)}{2}$, $\cos^2(x) = \frac{1 + \cos(2x)}{2}$)
• Examples of trigonometric integrals:
  • $\int \sin^3(x)\cos^2(x)dx$
  • $\int \frac{dx}{\sin(x)}$
  • $\int \frac{dx}{1 + \tan^2(x)}$