Integration by Parts

Integration by Parts

Part 1 of 7 — The Formula

Integration by Parts Formula

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

LIATE Rule for Choosing $u$

Choose $u$ in this priority order:

1. Logarithmic ($ln x$, $log x$)
2. Inverse trig ($arctan x$, $arcsin x$)
3. Algebraic (, , polynomials)

Worked Example

$int x,e^x,dx$

$u = x$, $dv = e^x,dx$

$du = dx$, $v = e^x$

$int x,e^x,dx = xe^x - int e^x,dx = xe^x - e^x + C = e^x(x-1) + C$

Integration by Parts 🎯

Key Takeaways — Part 1

1. $\int u\,dv = uv - \int v\,du$
2. Use LIATE to choose $u$
3. $\int \ln xdx=x$

Integration by Parts

Part 2 of 7 — Tabular Method

The Tabular (Column) Method

For $int (\text{polynomial}) cdot (\text{easy to integrate}),dx$:

Integration by Parts

Part 3 of 7 — Cycling (Boomerang) Problems

When IBP Cycles Back

For $int e^x sin x,dx$, doing IBP twice brings back the original integral!

$I = int e^x sin x,dx$

Integration by Parts

Part 4 of 7 — Definite Integrals with IBP

Formula for Definite Integrals

$int_a^b u,dv = [uv]_a^b - int_a^b v,du$

Integration by Parts

Part 5 of 7 — Special Cases

Inverse Trig Integrals

$int arctan x,dx$: $u = arctan x$, $dv = dx$

Integration by Parts

Part 6 of 7 — Practice Workshop

Mixed IBP Practice 🎯

Workshop Complete!

Integration by Parts — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Integration by Parts — Complete! ✅