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Integration Techniques Guide

Master when and how to use different integration methods. Learn to recognize patterns and choose the right technique.

Create flashcards for substitution patterns with FlashForge and visualize your solution strategies using MindMapFlow before tackling a new integral.

🧭 Quick Decision Guide

Ask yourself:

  1. Is it a basic form? → Use power rule, exponential, or trig formulas directly
  2. Can I simplify first? → Factor, expand, or break apart the expression
  3. Is there a composite function? → Try u-substitution
  4. Is it a product of different types? → Try integration by parts
  5. Is there a rational function? → Use partial fractions
  6. Does it involve √(a² ± x²)? → Try trigonometric substitution

🔄 U-Substitution

When to use: The integrand contains a function and its derivative (or a multiple of it).

Pattern Recognition:

Look for: ∫ f(g(x)) · g'(x) dx

If you see an "outer function" applied to an "inner function" whose derivative is also present, use u-substitution.

✓ Good Candidates

  • ∫ 2x · e^(x²) dx
  • ∫ sin(x) · cos(x) dx
  • ∫ x/(x²+1) dx
  • ∫ (3x²)/(x³+5) dx

✗ Bad Candidates

  • ∫ x · e^x dx (use integration by parts)
  • ∫ e^x · sin(x) dx (use integration by parts)
  • ∫ 1/(x²+1) dx (direct arctan formula)

Example Walkthrough

∫ 2x · cos(x²) dx

Step 1: Let u = x²

Step 2: Then du = 2x dx (perfect match!)

Step 3: ∫ cos(u) du = sin(u) + C

Step 4: = sin(x²) + C

Try with Integral Calculator →

🔀 Integration by Parts

When to use: Product of two different types of functions (polynomial × trig, polynomial × exponential, etc.)

Formula:

∫ u dv = uv - ∫ v du

Choose u using LIATE priority:

  • Logarithmic (ln x, log x)
  • Inverse trig (arcsin x, arctan x)
  • Algebraic (x², x³, polynomials)
  • Trigonometric (sin x, cos x)
  • Exponential (e^x, 2^x)

✓ Good Candidates

  • ∫ x · e^x dx
  • ∫ x² · sin(x) dx
  • ∫ ln(x) dx
  • ∫ x · arctan(x) dx

✗ Bad Candidates

  • ∫ x · √(x²+1) dx (u-sub works better)
  • ∫ sin²(x) dx (use trig identity first)
  • ∫ x/(x²+1) dx (u-substitution is easier)

Example Walkthrough

∫ x · e^x dx

Step 1: u = x (algebraic), dv = e^x dx (exponential)

Step 2: du = dx, v = e^x

Step 3: = x·e^x - ∫ e^x dx

Step 4: = x·e^x - e^x + C = e^x(x-1) + C

🧩 Partial Fractions

When to use: Rational functions (polynomial divided by polynomial) where degree of numerator < degree of denominator.

Steps:

  1. Factor the denominator completely
  2. Set up partial fraction decomposition based on factor types
  3. Solve for unknown coefficients (A, B, C...)
  4. Integrate each simpler fraction

Example

∫ 1/((x-1)(x+2)) dx

Step 1: 1/((x-1)(x+2)) = A/(x-1) + B/(x+2)

Step 2: Solve: A = 1/3, B = -1/3

Step 3: = (1/3)∫ 1/(x-1) dx - (1/3)∫ 1/(x+2) dx

Step 4: = (1/3)ln|x-1| - (1/3)ln|x+2| + C

📐 Trigonometric Substitution

When to use: Integrals involving √(a² - x²), √(a² + x²), or √(x² - a²)

Substitution Guide:

√(a² - x²)→ x = a·sin(θ)
√(a² + x²)→ x = a·tan(θ)
√(x² - a²)→ x = a·sec(θ)

Why It Works

These substitutions use Pythagorean identities to eliminate the square root:

  • sin²(θ) + cos²(θ) = 1
  • 1 + tan²(θ) = sec²(θ)
  • sec²(θ) - 1 = tan²(θ)

⚡ Special Techniques

Completing the Square

When: Quadratics that don't factor nicely, especially with integrals leading to arctan or ln.

Example: ∫ 1/(x² + 6x + 13) dx → Complete square: (x+3)² + 4 → arctan formula

Trigonometric Identities

When: Powers of sin and cos, or products like sin(mx)cos(nx).

Useful: sin²(x) = (1-cos(2x))/2, cos²(x) = (1+cos(2x))/2

Long Division

When: Rational function where numerator degree ≥ denominator degree.

Example: ∫ (x³+1)/(x²) dx → Divide first → ∫ (x + 1/x²) dx

🎯 Practice Strategy

To master integration techniques:

  1. Pattern recognition is key: Practice identifying which technique to use BEFORE starting.
  2. Sometimes multiple techniques work: Choose the easiest one (usually u-substitution if applicable).
  3. Check your work: Take the derivative of your answer. It should match the integrand.
  4. Build intuition: After solving, reflect on WHY that technique worked for that problem.
  5. Don't forget algebra: Many integrals require algebraic simplification first.

💡 Tip: Working through practice problems? Use our integral calculator for quick calculations while you focus on technique selection!

Ready to Practice?

Apply these techniques with our integral calculator and see step-by-step solutions.

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