Integration Techniques

Why Techniques Are Needed

Substitution (u-Substitution)

Notation Inside the Techniques

Integration by Parts

Trigonometric Integrals

Trigonometric Substitution

Partial Fractions

Choosing the Right Technique

Summary: The Techniques at a Glance

Integration Techniques FAQ

Beyond Direct Formulas

Most functions resist direct antidifferentiation. The integral of

e^{x^2}

has no elementary formula. Even innocuous-looking expressions like

\sqrt{1 + x^3}

lack closed-form antiderivatives.

Integration techniques transform difficult integrals into tractable ones. Substitution reverses the chain rule. Integration by parts reverses the product rule. Partial fractions decompose rational functions. Trigonometric methods handle roots and powers.

No single algorithm covers all cases—unlike differentiation, which follows systematic rules. Success requires recognizing which technique applies and executing it correctly. This pattern-matching skill develops through practice.

Key Terms

Integrand— the expression being transformed into a recognizable form

Indefinite Integral— the target: find the antiderivative family

Definite Integral— substitution in definite integrals requires converting bounds

See All Calculus Definitions →

Substitution (u-Substitution)

Substitution reverses the chain rule. If the integrand contains a function and its derivative, substitution simplifies.

Method: Let

u = g(x)

, so

du = g'(x)\, dx

. Replace all

x

-expressions with

u

-expressions and integrate.

Substitution Rule

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

Learn more about this formula: Substitution Rule →

Browse all integral formulas →

Example:

\int 2x \cos(x^2)\, dx

Let

u = x^2

, so

du = 2x\, dx

= \int \cos u\, du = \sin u + C = \sin(x^2) + C

For definite integrals, convert the limits: when

x = a

u = g(a)

; when

x = b

u = g(b)

Notation Inside the Techniques

Notation

Notation Inside the Techniques

Substitution's paired equations, the four-slot ledger of parts, and the substitution that runs backwards — the bookkeeping is the technique.

The integral signs — indefinite and definite; the algebra of

du

— differentials.

u = g(x)

\; du = g'(x)\,dx

Let u equal g of x, so d u equals g prime of x, d x

A change of name recorded as two equations — and the second is real differential algebra, not ritual. Everything must convert: integrand,

dx

, all of it, as in Substitution above.

CasesIn a definite integral the bounds convert too, and the transition is often labelled on the sign itself:

\int_{x=a}^{x=b}

becomes

\int_{u=g(a)}^{u=g(b)}

— after which there is no going back to

x

. The alternative workflow keeps the

x

-bounds and back-substitutes

u = g(x)

before evaluating; both are legal, mixing them is not.

Also written

w

t

when

u

is spoken for; the final return to

x

is the “back-substitution” step.

Do not confuseA partial conversion.

\int \cos u \,dx

— one variable in the integrand, another in the differential — is meaningless; a leftover

x

under a

du

means the substitution was not completed.

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

The integral of u d v equals u v minus the integral of v d u

Parts factors the integrand into two slots:

u

, destined for differentiation, and

dv

, destined for integration — and

dv

swallows the

dx

. The four-slot ledger

u, du, dv, v

organises the whole computation in Integration by Parts below.

CasesChoosing

u

follows LIATE — logarithmic, inverse trig, algebraic, trigonometric, exponential — a classroom mnemonic, not a theorem; it fails often enough to keep judgment in the loop.

Also written

\int u\,v'\,dx = uv - \int u'\,v\,dx

— the prime form, standard in European texts, where the differential never appears bare.

Do not confuseSubstitution's

u

. Same letter, different job: substitution's

u

renames the variable; parts'

u

labels a factor and

x

remains the variable throughout. Writing

dv

without its

dx

is the commonest ledger slip.

x = a\sin\theta

Let x equal a sine theta

Substitution running backwards: the old variable is expressed through the new one, opposite in direction to

u = g(x)

. The radical collapses through a Pythagorean identity, as in Trigonometric Substitution below.

CasesThree radical shapes, three choices:

\sqrt{a^2 - x^2} \to x = a\sin\theta

;

\sqrt{a^2 + x^2} \to x = a\tan\theta

;

\sqrt{x^2 - a^2} \to x = a\sec\theta

. The answer returns to

x

through

\theta = \arcsin\frac{x}{a}

and a reference triangle.

Also writtenHyperbolic versions —

x = a\sinh t

— in analysis and European texts, trading the identity

1 - \sin^2 = \cos^2

for

\cosh^2 - \sinh^2 = 1

Do not confuseThe

u

-direction. Here the new variable is not a function of

x

you can read off — inverting requires the inverse trig functions, and the domain restriction on

\theta

is what makes the inversion legal.

All calculus symbols →↑ Integrals

Integration by Parts

Integration by parts reverses the product rule:

Integration by Parts

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

Learn more about this formula: Integration by Parts →

Browse all integral formulas →

Method: Identify factors

u

and

dv

in the integrand. Differentiate

u

to get

du

; integrate

dv

to get

v

. Apply the formula.

Example:

\int x e^x\, dx

Let

u = x

and

dv = e^x\, dx

. Then

du = dx

and

v = e^x

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

Choosing the first factor: LIATE guides selection—Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Earlier types typically make better choices for

u

Trigonometric Integrals

Integrals involving powers of sine and cosine require strategic use of identities.

Odd power of sine: Save one

\sin x

, convert remaining

\sin^2 x = 1 - \cos^2 x

, substitute

u = \cos x

.

Odd power of cosine: Save one

\cos x

, convert remaining

\cos^2 x = 1 - \sin^2 x

, substitute

u = \sin x

.

Both powers even: Use half-angle identities:

\sin^2 x = \frac{1 - \cos 2x}{2} \qquad \cos^2 x = \frac{1 + \cos 2x}{2}

Example:

\int \sin^3 x\, dx = \int \sin x (1 - \cos^2 x)\, dx

Let

u = \cos x

= -\int (1 - u^2)\, du = -u + \frac{u^3}{3} + C = -\cos x + \frac{\cos^3 x}{3} + C

Power pattern	Strategy	Substitution
Odd power of sin x	save one sin x, rewrite the rest using sin²x = 1 − cos²x	u = cos x, du = −sin x dx
Odd power of cos x	save one cos x, rewrite the rest using cos²x = 1 − sin²x	u = sin x, du = cos x dx
Both powers even	apply half-angle identities to lower the powers	none directly — reduce, then integrate term by term

Trigonometric Substitution

Square roots of quadratics suggest trigonometric substitutions.

For

\sqrt{a^2 - x^2}

: Let

x = a\sin\theta

, so

\sqrt{a^2 - x^2} = a\cos\theta

For

\sqrt{a^2 + x^2}

: Let

x = a\tan\theta

, so

\sqrt{a^2 + x^2} = a\sec\theta

For

\sqrt{x^2 - a^2}

: Let

x = a\sec\theta

, so

\sqrt{x^2 - a^2} = a\tan\theta

Example:

\int \frac{1}{\sqrt{1 - x^2}}\, dx

Let

x = \sin\theta

, so

dx = \cos\theta\, d\theta

and

\sqrt{1 - x^2} = \cos\theta

= \int \frac{\cos\theta}{\cos\theta}\, d\theta = \int d\theta = \theta + C = \arcsin x + C

Radical in integrand	Substitution	Radical simplifies to
√(a² − x²)	x = a sin θ	a cos θ
√(a² + x²)	x = a tan θ	a sec θ
√(x² − a²)	x = a sec θ	a tan θ

Partial Fractions

Rational functions—polynomials divided by polynomials—decompose into simpler fractions.

Method: Factor the denominator. Write the fraction as a sum of terms with linear or irreducible quadratic denominators. Solve for coefficients. Integrate each term.

Example:

\int \frac{1}{x^2 - 1}\, dx = \int \frac{1}{(x-1)(x+1)}\, dx

Decompose:

\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}

Solving gives

A = 1/2

B = -1/2

= \frac{1}{2}\int \frac{1}{x-1}\, dx - \frac{1}{2}\int \frac{1}{x+1}\, dx = \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C

Choosing the Right Technique

Pattern recognition guides technique selection.

Substitution: Look for a function paired with its derivative. Expressions like

f(g(x)) \cdot g'(x)

signal substitution with

u = g(x)

.

Parts: Products of different function types—polynomial times exponential, polynomial times trigonometric, logarithm times polynomial.

Trigonometric integrals: Powers of

\sin x

and

\cos x

, products of trigonometric functions.

Trigonometric substitution: Square roots of

a^2 - x^2

a^2 + x^2

, or

x^2 - a^2

.

Partial fractions: Rational functions with factorable denominators.

Multiple techniques often combine. A problem might require substitution followed by parts, or partial fractions followed by a trigonometric integral.

Summary: The Techniques at a Glance

The techniques covered above all share the same goal—reshape the integrand until it matches a known formula—but each one reverses a different differentiation rule and is triggered by a different signal in the integrand. The table below collects all five in one place, pairing each technique with what it reverses, the pattern that signals it, the standard setup, and a canonical example. Read the "Recognize when…" column first when scanning an unfamiliar integral; read across the row once a match is found.

Technique	What it reverses	Recognize when…	Setup	Example trigger
u-substitution	chain rule	integrand contains a function and its derivative	u = g(x), du = g'(x) dx; rewrite and integrate in u	∫ 2x cos(x²) dx
Integration by parts	product rule	product of unrelated factors (poly · exp, poly · trig, log · poly)	∫ u dv = uv − ∫ v du; LIATE picks u	∫ x eˣ dx
Trigonometric integrals	Pythagorean & half-angle identities	powers and products of sin x, cos x (and tan, sec variants)	save a factor; convert via identity; substitute u = sin x or cos x	∫ sin³ x dx
Trigonometric substitution	Pythagorean identity	integrand contains √(a² − x²), √(a² + x²), or √(x² − a²)	x = a sin θ, a tan θ, or a sec θ depending on the radical	∫ 1 / √(1 − x²) dx
Partial fractions	addition of rational fractions	rational function with a factorable denominator	decompose into A/(x − r) + B/(x − s) + … and integrate each piece	∫ 1 / (x² − 1) dx

Integration Techniques FAQ

Do you have to change the limits when you use u-substitution?

You have two legal options, and the only mistake is mixing them. Either convert the bounds — when x = a, u = g(a) — and finish entirely in u, never returning to x; or keep the original x-bounds, back-substitute u = g(x) to get the answer in x, and only then evaluate. Evaluating u-bounds against an x-expression is the error.Read more →

What does it mean if there's a leftover x after substituting?

It means the substitution is incomplete, and the expression is meaningless as written. Something like ∫cos u dx mixes one variable in the integrand with another in the differential. Every x must convert, the integrand and the dx alike. A stray x usually signals that the chosen u does not have its derivative present.Read more →

Is the u in integration by parts the same as the u in u-substitution?

No — same letter, different job. In substitution, u renames the variable: everything converts and x disappears. In integration by parts, u labels one factor of the integrand and x remains the variable throughout; the ledger is u, du, dv, v. Reading a parts problem as a substitution, or the reverse, is a common source of stuck work.Read more →

Does LIATE always work?

No. LIATE — logarithmic, inverse trig, algebraic, trigonometric, exponential — is a classroom mnemonic for choosing u in integration by parts, not a theorem. It picks well most of the time, but it fails often enough that judgment stays in the loop. If the resulting ∫v du is harder than what you started with, swap the roles.Read more →

Why is trigonometric substitution written x = a sin θ instead of u = g(x)?

Because it runs backwards. Ordinary substitution names the new variable as a function of the old one; trigonometric substitution expresses the old variable through the new one, which is what lets a Pythagorean identity collapse the radical. The price is that returning to x needs an inverse trig function, a reference triangle, and a domain restriction on θ.Read more →