How do I know which method to use in integration?

Step 1: Identify the type of integral

Look at the expression first:

Case 1: Simple basic functions

If the integral looks like:

• xⁿ

• sinx, cosx

• eˣ

  Use basic integration rules

Case 2: Product of two functions

If you see:

• x·eˣ
• x·sinx
• x·lnx

Use Integration by Parts

Formula:
∫ u dv = uv − ∫ v du

Case 3: One function inside another

If you see something like:

• sin(2x)
• (3x + 1)⁵
• e^(x²)

Use Substitution (u-substitution)
Let u = inner function

Case 4: Rational functions

If you see:

• (x + 1)/(x² − 1)
• 1/(x² + 3x + 2)

Use Partial Fractions

Case 5: Trigonometric identities needed

If you see:

• sin²x, cos²x
• products like sinx cosx

Use Trig identities first

Step 2: Quick decision rule

Ask yourself:

• Can I simplify directly? → Basic rules
• Is it product? → Parts
• Is there a “function inside function”? → Substitution
• Is it a fraction of polynomials? → Partial fractions
• Do trig powers appear? → Identities first

Final Tip

Most integration problems become obvious after practice - pattern recognition is key, not memorization.