Integration Rules and Computation

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1Introduction

This lecture derives integration rules from differentiation rules and explains how the structure of an integrand determines whether direct formulas, substitution, or integration by parts is appropriate. An indefinite integral is a set of antiderivatives, and the fundamental theorem of calculus connects that set to the evaluation of definite integrals.

2Linearity and Constants of Integration

If $F^{'} = f$ and $G^{'} = g$ , then $(a F + b G)^{'} = a f + b g$ . Consequently, the set of antiderivatives of $a f + b g$ is

\int (a f (x) + b g (x)) d x = {a F (x) + b G (x) + C ∣ C \in R} .

An indefinite-integral identity represents a set of antiderivatives that differ by constants, not one particular function.

3Direct Formulas

Each integration formula can be verified by differentiating its right-hand side. Let $n$ be an integer other than $- 1$ . When $n < 0$ , the formula holds on intervals where $x \neq 0$ :

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C .

The exceptional exponent $n = - 1$ has the separate formula

\int \frac{1}{x} d x = \ln | x | + C .

This formula holds on each interval that excludes $0$ ; it does not define one antiderivative across that singular point.

For basic exponential and trigonometric functions,

\int e^{x} d x = e^{x} + C, \int \sin x d x = - \cos x + C, \int \cos x d x = \sin x + C .

If $a > 0$ and $a \neq 1$ , then

\int a^{x} d x = \frac{a^{x}}{\ln a} + C .

Applying a formula requires checking both the corresponding differentiation rule and the domain of the function.

4Substitution

Substitution reverses the chain rule. If $F^{'} = f$ and $g$ is differentiable, then

\frac{d}{d x} F (g (x)) = f (g (x)) g^{'} (x),

\int f (g (x)) g^{'} (x) d x = F (g (x)) + C .

The identifying feature is the simultaneous presence of a composite function $f (g (x))$ and the derivative $g^{'} (x)$ of its inner function.

For example, let $u = x^{2}$ , so that $d u = 2 x d x$ . Then

\int 2 x \cos (x^{2}) d x = \int \cos u d u = \sin (x^{2}) + C .

Differentiation of the result recovers the original integrand, including the factor $2 x$ .

For a definite integral, the endpoints must also be transformed. If $g$ is continuously differentiable on $[a, b]$ , meaning that $g^{'}$ is continuous, and $f$ is continuous on an interval containing $g ([a, b])$ , then

\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u .

For $p < q$ , define $\int_{q}^{p} f (u) d u = - \int_{p}^{q} f (u) d u$ . The same substitution formula therefore applies when $g (a) > g (b)$ .

5Integration by Parts

Integration by parts reverses the product rule. Integrating and rearranging $(u v)^{'} = u^{'} v + u v^{'}$ gives

\int u (x) v^{'} (x) d x = u (x) v (x) - \int u^{'} (x) v (x) d x .

If $u^{'}$ and $v^{'}$ are continuous on $[a, b]$ , then for definite integrals,

\int_{a}^{b} u (x) v^{'} (x) d x = [u (x) v (x)]_{a^{b}} - \int_{a}^{b} u^{'} (x) v (x) d x .

This method is useful when differentiating one factor simplifies it and the other factor has an accessible antiderivative. Taking $u = x$ and $v^{'} = e^{x}$ yields

\int x e^{x} d x = x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + C .

For a product such as $x^{m} e^{x}$ with $m$ a nonnegative integer, the method can be repeated until the polynomial degree reaches zero.

6Decomposition of Rational Functions

A rational function is a quotient of polynomials. If the numerator degree is at least the denominator degree, apply polynomial division first. If the denominator of the resulting proper fraction can be factored, partial-fraction decomposition can reduce the integral to direct formulas. For example, when $x \neq \pm 1$ ,

\frac{1}{x^{2} - 1} = \frac{1}{2} (\frac{1}{x - 1} - \frac{1}{x + 1}) .

Hence, on each interval that excludes $1$ and $- 1$ ,

\int \frac{1}{x^{2} - 1} d x = \frac{1}{2} \ln | x - 1 | - \frac{1}{2} \ln | x + 1 | + C .

7Selection by Structure

Structure of the integrand	First candidate	Basis for the choice
Basic functions or their linear combinations	Direct formulas	Known differentiation rules
$f (g (x)) g^{'} (x)$	Substitution	Chain rule
$g^{'} (x) / g (x)$	Substitution	$(\ln \| g \|)^{'} = g^{'} / g$
Product of different types of functions	Integration by parts	Simplification after differentiating one factor
Rational function	Division and partial fractions	Algebraic decomposition

This table is not a mechanical priority list. One must check whether the transformed integral is simpler and combine methods when necessary.

8Conditions and Limitations

In substitution, account exactly for the factor corresponding to $g^{'} (x)$ and any constant multiple, and transform the endpoints of a definite integral. In integration by parts, do not omit the boundary term $[u v]_{a^{b}}$ . For an integrand with a singularity, such as $1 / x$ , an ordinary definite-integral interval must exclude the singular point. An integral with a singularity at an endpoint or in the interior lies outside the ordinary definite integrals treated in this lecture.

The formulas derived in this lecture do not immediately reduce an antiderivative of every continuous function to previously known expressions. For example, directly applying the three methods to $e^{- x^{2}}$ does not reduce it to the basic formulas presented so far. When a definite value is required, numerical integration is another option.

9Exercises

data/exercise/math/calculus/integration-methods-and-computation.exercise.n.md data/exercise/math/calculus/substitution-and-integration-by-parts.exercise.n.md

10Related Material

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