January 22, 2025 • 12 min read

How to Solve Any Integral: A Complete Strategy Guide

Staring at an integral and not knowing where to start? This systematic approach will help you tackle any integration problem with confidence.

The Integration Mindset

Unlike differentiation, which follows clear mechanical rules, integration requires strategy and pattern recognition. There's no single algorithm that solves every integral. But there IS a systematic approach that works for the vast majority of problems you'll encounter.

Think of integration like a decision tree: you analyze the integral, identify patterns, and choose the appropriate technique. With practice, this becomes second nature.

Step 1: Can You Integrate Directly?

Before trying any fancy techniques, always check if you can apply basic formulas directly.

Ask Yourself:

  • Is this a power function? → Power Rule
  • Is this eˣ or aˣ? → Exponential Rule
  • Is this 1/x? → Natural Log
  • Is this a basic trig function? → Trig Formulas

Example:

$$\int 5x^3 + 2x - 7 \, dx = \frac{5x^4}{4} + x^2 - 7x + C$$

This one's easy—just apply basic rules term by term.

Step 2: Simplify First

Many integrals that look complicated become simple after algebraic manipulation.

Common Simplifications:

  • Expand products: (x+1)² → x² + 2x + 1
  • Divide fractions: (x² + x)/x → x + 1
  • Factor and cancel: Reduce before integrating
  • Use trig identities: sin²x = (1-cos2x)/2
  • Rewrite radicals: √x = x^(1/2)

Example:

$$\int \frac{x^3 + x^2}{x} \, dx = \int (x^2 + x) \, dx = \frac{x^3}{3} + \frac{x^2}{2} + C$$

Step 3: Look for Substitution Patterns

U-substitution is the most common technique. Look for these patterns:

The Key Pattern

$$\int f(g(x)) \cdot g'(x) \, dx$$

If you see a function composed with another, and the derivative of the inner function is present, use substitution!

How to Spot Substitution Opportunities:

  1. Look for "inside" functions — expressions inside parentheses, under radicals, or in exponents
  2. Check if the derivative is nearby — is the derivative of the inside function (or a constant multiple) also in the integral?
  3. Let u = inside function — and see if everything converts cleanly

Example:

$$\int 2x \cdot e^{x^2} \, dx$$
  • Inside function: x²
  • Its derivative 2x is right there!
  • Let u = x², du = 2x dx
  • ∫ eᵘ du = eᵘ + C = e^(x²) + C

Step 4: Consider Integration by Parts

When you have a product of two different types of functions that substitution can't handle, try integration by parts.

Integration by Parts

$$\int u \, dv = uv - \int v \, du$$

Use LIATE to Choose u:

  • Logarithmic (ln x, log x)
  • Inverse trig (arcsin, arctan)
  • Algebraic (x, x², polynomials)
  • Trigonometric (sin, cos)
  • Exponential (eˣ)

Functions earlier in LIATE should typically be u.

Example: ∫ x·sin(x) dx

  • u = x (algebraic), dv = sin(x) dx
  • du = dx, v = -cos(x)
  • = -x·cos(x) + ∫ cos(x) dx
  • = -x·cos(x) + sin(x) + C

Step 5: Try Partial Fractions for Rational Functions

If you have a fraction with polynomials (rational function), partial fraction decomposition often works.

When to Use Partial Fractions:

  • Integrand is P(x)/Q(x) where both are polynomials
  • Degree of numerator < degree of denominator
  • Denominator can be factored

Example:

$$\int \frac{1}{x^2-1} \, dx = \int \frac{1}{(x-1)(x+1)} \, dx$$

Decompose: 1/[(x-1)(x+1)] = A/(x-1) + B/(x+1)

Solving: A = 1/2, B = -1/2

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

Step 6: Trig Substitution for Square Roots

When you see √(a²-x²), √(a²+x²), or √(x²-a²), trig substitution is your friend.

The Three Cases:

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

The Complete Decision Tree

  1. Direct integration possible? → Apply basic formulas
  2. Can you simplify? → Algebra, trig identities, rewrite
  3. See f(g(x))·g'(x) pattern? → U-substitution
  4. Product of different function types? → Integration by parts
  5. Rational function? → Partial fractions
  6. Square root of quadratic? → Trig substitution
  7. Powers of trig functions? → Trig identities + reduction
  8. Nothing works? → Tables, CAS, or accept no elementary form

Practice Makes Perfect

The key to mastering integration is practice. The more integrals you solve, the faster you'll recognize patterns. Start with our practice problems:

Practice Problems Review Techniques