Techniques of Integration

Techniques of Integration

$Learning Objectives$

Select an integration technique from the algebraic structure of the integrand.
Apply substitution and integration by parts with mathematically valid choices.
Integrate powers and products of trigonometric functions using parity-aware identities.
Use trigonometric substitutions with explicit domain and sign assumptions.
Decompose rational functions into partial fractions and integrate the resulting terms.
Apply Wallis' formula, reduction formulas, hyperbolic identities, and Gamma/Beta functions where appropriate.
Verify indefinite integrals by differentiation.

Integration techniques are transformations that convert a difficult integral into one or more standard integrals. No single mnemonic determines the correct method in every case; the structure of the integrand, available derivatives, algebraic identities, and domain assumptions all matter.

$Verification Standard$

For an indefinite integral, a proposed antiderivative $F(x)+C$ is correct on an interval if $F'(x)$ equals the original integrand throughout that interval. Differentiating the final result is the most direct verification.

Integration by Substitution

$Substitution Rule$

If $u=g(x)$ and $du=g'(x)\,dx$ , then

\int f(g(x))g'(x)\,dx=\int f(u)\,du.

The method reverses the chain rule.

$Choosing and Applying a Substitution$

STEP-BY-STEP

Look for a repeated inner expression $g(x)$ whose derivative is also present up to a constant factor.
Set $u=g(x)$ and compute $du=g'(x)\,dx$ .
Rewrite the entire integrand in terms of $u$ ; do not leave a mixture of $x$ and $u$ .
Integrate in $u$ and back-substitute for indefinite integrals.
For definite integrals, either change the bounds to $u$ -values immediately or back-substitute before evaluating; do not mix the two procedures.

$Rationalizing Substitution$

For radicals such as $\sqrt[n]{ax+b}$ , a substitution like $u=\sqrt[n]{ax+b}$ can eliminate the fractional power because $u^n=ax+b$ and $n u^{n-1}du=a\,dx$ .

Integration by Parts

Reverse form of the product rule.

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

$LIATE Is a Heuristic, Not a Theorem$

LIATE—Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential—is a useful priority guide for choosing $u$ , but it does not guarantee the simplest path and it does not make other valid choices mathematically wrong. A good choice typically makes $du$ simpler while leaving $dv$ easy to integrate.

$Integration by Parts Workflow$

STEP-BY-STEP

Split the integrand into a differentiable factor $u$ and an integrable factor $dv$ .
Compute $du$ and $v$ .
Apply $\int u\,dv=uv-\int v\,du$ .
Check whether the new integral is simpler, whether repeated integration by parts is useful, or whether a cyclic equation appears.
Differentiate the final antiderivative to verify it.

Tabular Integration by Parts

The tabular method is efficient when one factor differentiates to zero after finitely many derivatives, commonly a polynomial, while the other factor can be integrated repeatedly, commonly an exponential or sine/cosine function.

$Tabular Method$

STEP-BY-STEP

Differentiate the polynomial column until zero.
Repeatedly integrate the other factor.
Apply alternating signs $+,-,+,-,\ldots$ .
Multiply diagonal pairs and sum them.
Differentiate the result as a final check.

$Interactive Simulation$

The simulation treats LIATE as guidance and distinguishes a convenient choice from another mathematically valid choice.

Integration by Parts Choice Explorer

\int x e^x\,dx

LIATE is a heuristic for convenience, not a theorem that makes every other split wrong.

Compare valid splitsLIATE-preferred: Recommended
Differentiating x simplifies immediately.Alternative: Valid, but less direct
The formula remains valid, but the new integral is harder.

u=x,\qquad dv=e^x\,dx

Recommended. Differentiating x simplifies immediately.

Reference result and derivative check

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

\frac{d}{dx}[e^x(x-1)]=xe^x

Reduction Formulas

$Examples of Reduction Formulas$

For integer $n\ge2$ ,

\int \sin^n x\,dx=-\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int\sin^{n-2}x\,dx,

\int \cos^n x\,dx=\frac{\cos^{n-1}x\sin x}{n}+\frac{n-1}{n}\int\cos^{n-2}x\,dx.

These identities reduce the exponent by two and are obtained by integration by parts.

Trigonometric Integrals

Powers of Sine and Cosine

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

STEP-BY-STEP

If $m$ is odd and positive, save one $\sin x$ , convert the remaining even sine power using $\sin^2x=1-\cos^2x$ , and use $u=\cos x$ .
If $n$ is odd and positive, save one $\cos x$ , convert the remaining even cosine power using $\cos^2x=1-\sin^2x$ , and use $u=\sin x$ .
If both powers are even, use power-reduction identities such as $\sin^2x=(1-\cos2x)/2$ and $\cos^2x=(1+\cos2x)/2$ .

Powers of Tangent and Secant

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

STEP-BY-STEP

If $n$ is even and $n\ge2$ , save one factor $\sec^2x\,dx$ , convert the remaining even secant power with $\sec^2x=1+\tan^2x$ , and use $u=\tan x$ .
If $m$ is odd, $m\ge1$ , and at least one factor of $\sec x$ is present, save $\sec x\tan x\,dx$ , convert the remaining even tangent power with $\tan^2x=\sec^2x-1$ , and use $u=\sec x$ .
If the required differential factor is not actually present, do not invent it. For example, a pure odd power of secant generally requires a reduction formula or integration by parts rather than the $u=\sec x$ shortcut.

$The Choice Depends on Both Powers$

A statement such as “odd tangent power implies save $\sec x\tan x$ ” is incomplete unless a secant factor is available. Always inspect the full product before selecting the saved factor.

Trigonometric Substitution

Trigonometric substitutions transform quadratic radicals by using Pythagorean identities. Because $\sqrt{z^2}=|z|$ , the usual simplifications require an explicit angle interval or sign branch.

$Three Standard Forms with Sign Conditions$

Assume $a>0$ .

For $\sqrt{a^2-x^2}$ , use $x=a\sin\theta$ with $\theta\in[-\pi/2,\pi/2]$ . Then $\cos\theta\ge0$ and $\sqrt{a^2-x^2}=a\cos\theta$ .
For $\sqrt{a^2+x^2}$ , use $x=a\tan\theta$ with $\theta\in(-\pi/2,\pi/2)$ . Then $\sec\theta>0$ and $\sqrt{a^2+x^2}=a\sec\theta$ .
For $\sqrt{x^2-a^2}$ on the branch $x\ge a$ , use $x=a\sec\theta$ with $\theta\in[0,\pi/2)$ . Then $\tan\theta\ge0$ and $\sqrt{x^2-a^2}=a\tan\theta$ . A negative- $x$ branch requires a corresponding sign-aware choice.

$Interactive Simulation$

The simulation displays the absolute-value step first, then shows which interval assumption permits the simplified radical.

Trigonometric Substitution: Domains and Signs

Substitution

\sqrt{a^2-x^2}

x=a\sin\theta

Sign-safe simplification

\sqrt{a^2\cos^2\theta}=a|\cos\theta|

\theta\in[-\pi/2,\pi/2]

\Longrightarrow\ a\cos\theta

Reference triangle

opposite = x, hypotenuse = a, adjacent = √(a²−x²)

The absolute-value line is essential because √(z²)=|z|. The chosen angle interval is what justifies removing the absolute value.

Weierstrass Substitution

$Tangent Half-Angle Substitution$

With $t=\tan(x/2)$ ,

\sin x=\frac{2t}{1+t^2},\qquad \cos x=\frac{1-t^2}{1+t^2},\qquad dx=\frac{2\,dt}{1+t^2}.

This converts any rational expression in $\sin x$ and $\cos x$ into a rational function of $t$ on intervals where the substitution is valid.

Partial Fraction Decomposition

Partial fractions applies to rational functions $P(x)/Q(x)$ . If $\deg P\ge\deg Q$ , perform polynomial long division before decomposing the proper rational remainder.

$Decomposition Patterns$

Distinct linear factor $(x-a)$ : use $A/(x-a)$ .
Repeated linear factor $(x-a)^k$ : include $A_1/(x-a)+\cdots+A_k/(x-a)^k$ .
Irreducible quadratic factor $q(x)$ : use a linear numerator $(Ax+B)/q(x)$ ; repeat the pattern for powers of $q(x)$ .

Partial Fraction Decomposition Explorer

Denominator structure Distinct linear factors Repeated linear factor Irreducible quadratic

1. Original rational function

\frac1{(x-1)(x+2)}

2. Required decomposition form

\frac A{x-1}+\frac B{x+2}

3. Clear denominators

1=A(x+2)+B(x-1)

4. Solve constants

A=\frac13,\ B=-\frac13

5. Decomposed rational expression

\frac1{3(x-1)}-\frac1{3(x+2)}

Recombination check

\frac{(x+2)-(x-1)}{3(x-1)(x+2)}=\frac1{(x-1)(x+2)}

Wallis' Formula

For the nonnegative integer powers

I_n=\int_0^{\pi/2}\sin^n x\,dx=\int_0^{\pi/2}\cos^n x\,dx,

Wallis' recurrence is

I_n=\frac{n-1}{n}I_{n-2},\qquad n\ge2,

with $I_0=\pi/2$ and $I_1=1$ .

$Closed Product Forms$

For even $n$ ,

I_n=\frac{(n-1)(n-3)\cdots1}{n(n-2)\cdots2}\frac{\pi}{2}.

For odd $n$ ,

I_n=\frac{(n-1)(n-3)\cdots2}{n(n-2)\cdots1}.

Wallis Formula Explorer

Power n = 4

I_4=\int_0^{\pi/2}\sin^{4}x\,dx

I_4=\frac{3}{4}I_2

I_0=\pi/2

Numerical value: 0.58904862

The edge cases are part of the formula: I₀ = π/2 and I₁ = 1. Higher powers follow the reduction Iₙ = (n−1)Iₙ₋₂/n.

Hyperbolic-Function Integrals

$Core Identities$

\cosh^2x-\sinh^2x=1,\qquad \frac{d}{dx}\sinh x=\cosh x,\qquad \frac{d}{dx}\cosh x=\sinh x.

The same structural reasoning used for trigonometric powers often applies to hyperbolic powers, with the sign change in the fundamental identity handled carefully.

Gamma and Beta Functions

$Gamma Function$

For $s>0$ ,

\Gamma(s)=\int_0^\infty x^{s-1}e^{-x}\,dx,

and $\Gamma(s+1)=s\Gamma(s)$ . Hence $\Gamma(n)=(n-1)!$ for positive integers $n$ .

$Beta Function$

For $p,q>0$ ,

B(p,q)=\int_0^1 t^{p-1}(1-t)^{q-1}\,dt =\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.

$Technique Selection Is Structural$

A mnemonic can suggest a method, but the decisive question is whether the transformed integral is valid and simpler. Check algebra, domain assumptions, and the derivative of the final antiderivative.

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Learning Objectives

Verification Standard

Integration by Substitution

Substitution Rule

Choosing and Applying a Substitution

Rationalizing Substitution

Integration by Parts

Integration by Parts

LIATE Is a Heuristic, Not a Theorem

Integration by Parts Workflow

Tabular Integration by Parts

Tabular Method

Interactive Simulation

Integration by Parts Choice Explorer

Reduction Formulas

Examples of Reduction Formulas

Trigonometric Integrals

Powers of Sine and Cosine

Strategy for ∫sin⁡mxcos⁡nx dx\int \sin^m x\cos^n x\,dx∫sinmxcosnxdx

Powers of Tangent and Secant

Strategy for ∫tan⁡mxsec⁡nx dx\int \tan^m x\sec^n x\,dx∫tanmxsecnxdx

The Choice Depends on Both Powers

Trigonometric Substitution

Three Standard Forms with Sign Conditions

Interactive Simulation

Trigonometric Substitution: Domains and Signs

Weierstrass Substitution

Tangent Half-Angle Substitution

Partial Fraction Decomposition

Decomposition Patterns

Partial Fraction Decomposition Explorer

Wallis' Formula

Closed Product Forms

Wallis Formula Explorer

Hyperbolic-Function Integrals

Core Identities

Gamma and Beta Functions

Gamma Function

Beta Function

Technique Selection Is Structural

$Learning Objectives$

$Verification Standard$

$Substitution Rule$

$Choosing and Applying a Substitution$

$Rationalizing Substitution$

$LIATE Is a Heuristic, Not a Theorem$

$Integration by Parts Workflow$

$Tabular Method$

$Interactive Simulation$

$Examples of Reduction Formulas$

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

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

$The Choice Depends on Both Powers$

$Three Standard Forms with Sign Conditions$

$Interactive Simulation$

$Tangent Half-Angle Substitution$

$Decomposition Patterns$

$Closed Product Forms$

$Core Identities$

$Gamma Function$

$Beta Function$

$Technique Selection Is Structural$