Topics · 16 formulas

Techniques of Integration

Techniques of Integration in CalcRef: 16 free calculus formulas covering substitution, parts, trig sub, partial fractions, and improper integrals. Each one is shown with its notation and the conditions it holds under, and links to a page with a worked example, so you can revise the topic or find the exact rule you need.

u-Substitution

\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)

The reverse of the chain rule. Identify an inner function u = g(x), compute du = g'(x) dx, and substitute.

Substitution Methods

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

Three standard trig substitutions for integrals involving square roots of quadratics. Each eliminates the radical using a Pythagorean identity.

Substitution Methods

Integration by Parts

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

The reverse of the product rule. Choose u and dv from the integrand using the LIATE mnemonic (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).

Integration by Parts

Tabular Method (Repeated IBP)

\text{Alternating signs: } (+)u \cdot v_1 - u' \cdot v_2 + u'' \cdot v_3 - \cdots

A shortcut for repeated integration by parts when one factor is a polynomial. Create a table of derivatives and antiderivatives with alternating signs.

Integration by Parts

Trig Sub: √(a²-x²)

\sqrt{a^2 - x^2}: \text{ let } x = a\sin\theta,\; dx = a\cos\theta\, d\theta

For integrals with √(a²-x²), substitute x = a sin θ. Then √(a²-x²) = a cos θ.

Conditions: -a ≤ x ≤ a, -π/2 ≤ θ ≤ π/2.

Trigonometric Methods

Trig Sub: √(a²+x²)

\sqrt{a^2 + x^2}: \text{ let } x = a\tan\theta,\; dx = a\sec^2\theta\, d\theta

For integrals with √(a²+x²), substitute x = a tan θ. Then √(a²+x²) = a sec θ.

Conditions: -π/2 < θ < π/2.

Trigonometric Methods

Trig Sub: √(x²-a²)

\sqrt{x^2 - a^2}: \text{ let } x = a\sec\theta,\; dx = a\sec\theta\tan\theta\, d\theta

For integrals with √(x²-a²), substitute x = a sec θ. Then √(x²-a²) = a tan θ.

Conditions: x ≥ a or x ≤ -a. 0 ≤ θ < π/2 or π ≤ θ < 3π/2.

Trigonometric Methods

Trig Integrals: sinᵐx cosⁿx

\int \sin^m x \cos^n x \, dx

Strategy: If m or n is odd, save one factor and convert the rest using sin²+cos²=1. If both even, use half-angle identities: sin²x = (1-cos2x)/2, cos²x = (1+cos2x)/2.

Trigonometric Methods

Trig Integrals: secᵐx tanⁿx

\int \sec^m x \tan^n x \, dx

Strategy: If n is odd, save sec x tan x and convert tan² = sec²-1. If m is even, save sec²x and convert sec² = 1+tan².

Trigonometric Methods

Partial Fractions: Distinct Linear

\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decompose a rational function with distinct linear factors in the denominator into simpler fractions that can be integrated individually.

Conditions: Degree of P(x) must be less than the degree of the denominator.

Partial Fractions

Partial Fractions: Repeated Linear

\frac{P(x)}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}

For repeated linear factors, include a term for each power of the factor up to its multiplicity.

Partial Fractions

Partial Fractions: Irreducible Quadratic

\frac{P(x)}{(x^2+bx+c)} = \frac{Ax+B}{x^2+bx+c}

For irreducible quadratic factors (discriminant < 0), the numerator is a linear expression Ax + B.

Partial Fractions

Improper Integral: Type I (Infinite Limit)

\int_a^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_a^t f(x)\, dx

When the upper (or lower) limit is infinite, replace it with a variable t and take the limit. If the limit is finite, the integral converges.

Improper Integrals

Improper Integral: Type II (Discontinuity)

\int_a^b f(x)\, dx = \lim_{t \to a^+} \int_t^b f(x)\, dx \text{ (if } f \text{ unbounded at } a\text{)}

When f has a discontinuity in [a,b], approach the discontinuity as a limit.

Improper Integrals

Comparison Test for Integrals

0 \leq f(x) \leq g(x): \int g \text{ conv.} \Rightarrow \int f \text{ conv.}; \quad \int f \text{ div.} \Rightarrow \int g \text{ div.}

If 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges. If ∫f diverges, then ∫g diverges.

Improper Integrals

p-Integral Test

\int_1^{\infty} \frac{1}{x^p}\, dx \text{ converges if } p > 1 \text{, diverges if } p \leq 1

A fundamental convergence result: the improper integral of 1/xᵖ from 1 to ∞ converges if and only if p > 1.

Improper Integrals

See all 16 Techniques of Integration formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.