Techniques of Integration formula sheet

16 rows · CalcRef · sample of the first 6 rows

Techniques of Integration
Formula	Statement	Conditions
u-Substitution	$\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)$	no restriction stated
Trigonometric Substitution (Overview)	$\sqrt{a^2 - x^2}: x = a\sin\theta \quad \sqrt{a^2 + x^2}: x = a\tan\theta \quad \sqrt{x^2 - a^2}: x = a\sec\theta$	no restriction stated
Integration by Parts	$\int u\, dv = uv - \int v\, du$	no restriction stated
Tabular Method (Repeated IBP)	$\text{Alternating signs: } (+)u \cdot v_1 - u' \cdot v_2 + u'' \cdot v_3 - \cdots$	no restriction stated
Trig Sub: √(a²-x²)	$\sqrt{a^2 - x^2}: \text{ let } x = a\sin\theta,\; dx = a\cos\theta\, d\theta$	-a ≤ x ≤ a, -π/2 ≤ θ ≤ π/2.
Trig Sub: √(a²+x²)	$\sqrt{a^2 + x^2}: \text{ let } x = a\tan\theta,\; dx = a\sec^2\theta\, d\theta$	-π/2 < θ < π/2.

The 16 Techniques of Integration formulas in the CalcRef library, every one of them free to read at /topics/techniques-integration.

CalcRef is an educational reference. Check every statement on this sheet against your own course notes, textbook or exam formula policy before relying on it in graded work.

Every formula on this sheet is also free on its own page — browse the full reference. Premium pays for the sheet, not the mathematics.