Improper and Numerical Integration
Cheat sheet.

The key rules, formulas and reminders from all 2 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Improper Integrals and Convergence

Tail $\int_1^\infty x^{-p}dx$ : $p>1$ . Near zero $\int_0^1x^{-p}dx$ : $p<1$ .

Split every interior singularity. Both pieces must converge. Symmetric cancellation defines a principal value, not ordinary convergence.

|I-T_n|\le\frac{M_2(b-a)^3}{12n^2},

|I-S_n|\le\frac{M_4(b-a)^5}{180n^4}.

Simpson needs even $n$ on a uniform grid. Error bounds require the stated derivative bounds; sample agreement alone is no guarantee.

2 reference blocks

Tail $\int_1^\infty x^{-p}dx$ : $p>1$ . Near zero $\int_0^1x^{-p}dx$ : $p<1$ .

Split every interior singularity. Both pieces must converge. Symmetric cancellation defines a principal value, not ordinary convergence.

2 reference blocks

|I-T_n|\le\frac{M_2(b-a)^3}{12n^2},

|I-S_n|\le\frac{M_4(b-a)^5}{180n^4}.

Simpson needs even $n$ on a uniform grid. Error bounds require the stated derivative bounds; sample agreement alone is no guarantee.