Advanced Integration Techniques

Part 1 of 7 — Choosing a Method

Integration Decision Tree

1. Basic? Power rule, trig, exponential → do it directly
2. Composite? $int f(g(x))g'(x),dx$ → u-substitution
3. Product of different types? → Integration by parts
4. Rational function? → Partial fractions
5. Trig powers? → Trig identities
6. Square root of quadratic? → Trig substitution (beyond BC, but good to know)

Choose the Method 🎯

Key Takeaways — Part 1

Recognize the pattern first. Choose the right technique.

Advanced Integration

Part 2 of 7 — Challenging u-Substitutions

Tricky u-Sub Examples

int rac{e^x}{1 + e^x},dx: let $u = 1 + e^x$

int rac{ln x}{x},dx: let $u = ln x$

$int xsqrt{x+1},dx$: let $u = x+1$, so $x = u - 1$

Completing the Square for u-Sub

int rac{dx}{x^2 + 4x + 8} = int rac{dx}{(x+2)^2 + 4}: let $u = x + 2$

u-Sub 🎯

Key Takeaways — Part 2

Look for the derivative of a function inside the integral. Complete the square when needed.

Advanced Integration

Part 3 of 7 — Combining Techniques

Integration by Parts + u-Sub

$int e^{sqrt{x}},dx$:

Step 1: $u = sqrt{x}$, $x = u^2$, $dx = 2u,du$

$= int e^u cdot 2u,du$

Step 2: Integration by parts: $= 2(ue^u - e^u) + C = 2e^{sqrt{x}}(sqrt{x} - 1) + C$

Combined Methods 🎯

Key Takeaways — Part 3

Some integrals need multiple techniques in sequence.

Advanced Integration

Part 4 of 7 — Improper Integrals Revisited

Type I: Infinite Limits

int_1^{infty} rac{1}{x^p},dx converges iff $p > 1$

Type II: Discontinuities

int_0^1 rac{1}{sqrt{x}},dx = lim_{a o 0^+}int_a^1 x^{-1/2},dx = lim_{a o 0^+} [2sqrt{x}]_a^1 = 2

Comparison Test for Integrals

If $0 leq f(x) leq g(x)$:

• $int g$ converges $implies$ $int f$ converges
• $int f$ diverges $implies$ $int g$ diverges

Improper 🎯

Key Takeaways — Part 4

$p$-test: $\int_1^{\infty} 1/x^p$ converges iff $p > 1$. Use comparison for harder integrals.

Advanced Integration

Part 5 of 7 — Tabular Integration

Tabular Method (Repeated Parts)

For $int x^n e^{ax},dx$ or $int x^n sin(ax),dx$:

$int x^3 e^x,dx$:

$= x^3 e^x - 3x^2 e^x + 6x e^x - 6e^x + C$

Tabular 🎯

Key Takeaways — Part 5

Tabular method speeds up repeated integration by parts.

Advanced Integration

Part 6 of 7 — Practice Workshop

Workshop 🎯

Workshop Complete!

Advanced Integration — Review

Part 7 of 7 — Final Assessment

Final 🎯

Advanced Integration — Complete! ✅