Indefinite Integrals

Indefinite integrals are the “reverse process” of differentiation. While derivatives take a function and produce its rate of change, indefinite integrals take a function (often a derivative) and produce a whole family of original functions whose derivative is that given function.

Because this chapter is about indefinite integrals, we focus on antiderivatives and their properties, not on areas or limits (those belong elsewhere in Integral Calculus).

Antiderivatives and the Constant of Integration

Let $f$ be a function. A function $F$ is called an antiderivative of $f$ on an interval if
$$
F'(x) = f(x)
$$
for every $x$ in that interval.

Indefinite integration is a notation for “the set of all antiderivatives” of $f$. We write
$$
\int f(x)\,dx = F(x) + C,
$$
where:

• $\int$ is the integral sign,
• $f(x)$ is the integrand,
• $dx$ is part of the notation indicating the variable of integration,
• $F(x)$ is any one antiderivative of $f(x)$,
• $C$ is an arbitrary constant, called the constant of integration.

Why do we need $C$? Because differentiation “loses” constant information:

• If $F(x)$ is an antiderivative of $f(x)$, then so is $F(x) + K$ for any constant $K$, since $\frac{d}{dx}[F(x)+K]=F'(x)=f(x)$.
• All these functions differ only by a constant, so the indefinite integral represents this entire family by adding $+C$.

Example:

• $\dfrac{d}{dx}(x^2)=2x$.
• So an antiderivative of $2x$ is $x^2$.
• But also $\dfrac{d}{dx}(x^2+5)=2x$ and $\dfrac{d}{dx}(x^2-100)=2x$.
• Therefore
  $$
  \int 2x\,dx = x^2 + C.
  $$

A quick check: to verify an antiderivative, just differentiate your answer. If you get the integrand back, it’s correct.

Basic Indefinite Integral Formulas

You should recognize these as the “reverse” of basic derivative rules. In what follows, $C$ stands for an arbitrary constant.

Power Rule for Indefinite Integrals

For any real $n \neq -1$,
$$
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C.
$$

This is the inverse of the derivative rule $\dfrac{d}{dx} x^{n+1} = (n+1)x^n$.

Examples:

• $\displaystyle \int x^3\,dx = \frac{x^4}{4} + C$.
• $\displaystyle \int \sqrt{x}\,dx = \int x^{1/2}\,dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$.
• $\displaystyle \int \frac{1}{x^4}\,dx = \int x^{-4}\,dx = \frac{x^{-3}}{-3} + C = -\frac{1}{3x^3} + C$.

Special case $n=-1$:

• $\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C$ (for $x\neq 0$).

Exponential and Logarithmic Formulas

Standard exponentials:

• $\displaystyle \int e^x\,dx = e^x + C$.
• $\displaystyle \int a^x\,dx = \frac{a^x}{\ln a} + C$ for $a>0$, $a\neq 1$.

Natural logarithm:

• $\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C$ (already noted).
• More generally, if $u(x)$ is a differentiable function and $u'(x)$ is present:
  $$
  \int \frac{u'(x)}{u(x)}\,dx = \ln|u(x)| + C.
  $$

Basic Trigonometric Indefinite Integrals

Using known derivatives of sine, cosine, etc., we get:

• $\displaystyle \int \cos x\,dx = \sin x + C$.
• $\displaystyle \int \sin x\,dx = -\cos x + C$.
• $\displaystyle \int \sec^2 x\,dx = \tan x + C$.
• $\displaystyle \int \csc^2 x\,dx = -\cot x + C$.
• $\displaystyle \int \sec x \tan x\,dx = \sec x + C$.
• $\displaystyle \int \csc x \cot x\,dx = -\csc x + C$.

These are simply the reverse operations of the corresponding derivative rules.

Linearity of the Indefinite Integral

Indefinite integration is linear, just like differentiation. For functions $f,g$ and constants $a,b$:

1. Sum rule:
   $$
   \int \bigl(f(x) + g(x)\bigr)\,dx = \int f(x)\,dx + \int g(x)\,dx.
   $$
2. Constant multiple rule:
   $$
   \int a\,f(x)\,dx = a \int f(x)\,dx.
   $$

These allow you to integrate term-by-term and factor out constants.

Example:
$$
\int (3x^2 - 4x + 7)\,dx
= 3\int x^2\,dx - 4\int x\,dx + 7\int 1\,dx
= 3\cdot\frac{x^3}{3} - 4\cdot\frac{x^2}{2} + 7x + C
= x^3 - 2x^2 + 7x + C.
$$

Basic Techniques of Indefinite Integration

More complicated integrals can rarely be done directly by inspection. Several general techniques are used to transform an integral into a simpler one that you know how to evaluate. Here we outline the most common ones used with indefinite integrals.

Substitution (Reverse Chain Rule)

Substitution is the main method for handling integrals involving compositions of functions. It undoes the chain rule for derivatives.

Informal idea: if you can rewrite part of the integrand as $u(x)$, and the rest looks like $u'(x)\,dx$, then
$$
\int f(u(x))\,u'(x)\,dx = \int f(u)\,du.
$$

Practical steps (in $x$-notation):

1. Choose a substitution $u = g(x)$ that simplifies the integrand.
2. Compute $du = g'(x)\,dx$ (or $dx$ in terms of $du$ and $x$).
3. Rewrite the entire integrand in terms of $u$ and $du$.
4. Integrate with respect to $u$.
5. Replace $u$ back by $g(x)$ to obtain an answer in $x$.

Example:
$$
\int 2x\cos(x^2)\,dx.
$$
Let $u = x^2$, so $du = 2x\,dx$. Then
$$
\int 2x\cos(x^2)\,dx = \int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C.
$$

Substitution is especially helpful when:

• The integrand contains a “function times its derivative,” e.g. $f(g(x))g'(x)$.
• You see a complicated expression (like $x^2+1$ or $\sqrt{1-x^2}$) and its derivative (or a constant multiple) is elsewhere in the integrand.

Integration by Parts (Reverse Product Rule)

Integration by parts undoes the product rule for derivatives. If $u(x)$ and $v(x)$ are differentiable, the product rule says
$$
(uv)' = u'v + uv'.
$$
Integrating both sides and rearranging gives the formula for integration by parts:
$$
\int u\,dv = uv - \int v\,du.
$$
Here $u=u(x)$, $v=v(x)$, $du=u'(x)\,dx$, and $dv=v'(x)\,dx$.

To use it for an indefinite integral:

1. Identify parts of the integrand as $u$ and $dv$.
2. Compute $du$ and integrate $dv$ to get $v$.
3. Apply the formula $\int u\,dv = uv - \int v\,du$.
4. Integrate the new integral $\int v\,du$ if possible.
5. Add the constant of integration at the end.

Example:
$$
\int x e^x\,dx.
$$
Choose $u = x$, $dv = e^x\,dx$. Then

• $du = dx$,
• $v = e^x$.

So
$$
\int x e^x\,dx = x e^x - \int e^x \cdot 1\,dx
= x e^x - e^x + C
= e^x(x-1) + C.
$$

Integration by parts is especially useful for:

• Products of a polynomial with an exponential or trigonometric function ($x e^x$, $x\sin x$, etc.).
• Products involving $\ln x$ or inverse trigonometric functions (which are usually chosen as $u$).

Trigonometric Integrals (Overview)

Certain integrals of products or powers of $\sin x$, $\cos x$, and other trig functions can be simplified using identities. Details are typically organized as their own topic, but at a high level:

• Integrals of $\sin^m x\cos^n x$ often use identities such as
• $\sin^2 x = 1 - \cos^2 x$,
• $\cos^2 x = 1 - \sin^2 x$,
• $\sin(2x) = 2\sin x\cos x$.
• When one power is odd, substitution using $\sin^2x + \cos^2x = 1$ is common.
• Similar ideas apply to $\tan x$ and $\sec x$ with identities like $\tan^2x + 1 = \sec^2x$.

Here the important idea is that indefinite integrals involving trigonometric functions often require algebraic manipulation plus substitution to fit known basic forms.

Trigonometric Substitution (Overview)

Trigonometric substitution is used for integrals containing square roots of quadratic expressions in $x$, like:

• $\sqrt{a^2 - x^2}$,
• $\sqrt{a^2 + x^2}$,
• $\sqrt{x^2 - a^2}$.

The idea is to choose $x$ as a trig function so that the square root simplifies using identities like $\sin^2\theta + \cos^2\theta = 1$ or $\tan^2\theta + 1 = \sec^2\theta$. For example, for $\sqrt{a^2 - x^2}$ one often sets $x = a\sin\theta$.

You then:

1. Replace $x$ with a trig expression (e.g. $a\sin\theta$).
2. Rewrite $dx$ in terms of $d\theta$.
3. Simplify the integrand to something involving trig functions.
4. Integrate with respect to $\theta$.
5. Convert back to $x$ using the triangle or the original substitution.

While the details are more involved, trigonometric substitution is still just a method to transform an indefinite integral into one you know how to evaluate.

Partial Fractions (Overview)

When integrating rational functions (ratios of polynomials), a key technique is partial fraction decomposition.

Idea: A rational function
$$
\frac{P(x)}{Q(x)}
$$
is rewritten as a sum of simpler fractions, like
$$
\frac{A}{x-a},\quad \frac{Bx + C}{x^2+1},
$$
etc., which have standard antiderivatives such as $\ln|x-a|$ or $\arctan x$.

The general strategy is:

1. Factor the denominator $Q(x)$ as much as possible.
2. Write $\dfrac{P(x)}{Q(x)}$ as a sum of fractions with unknown constants in numerators (depending on the factor types).
3. Solve for these constants.
4. Integrate term-by-term using basic rules and (typically) substitution.

Again, the end result is an antiderivative plus $C$.

Indefinite Integrals and Families of Solutions

Because indefinite integrals give general antiderivatives, they are used to solve basic first-order differential equations of the form
$$
y' = f(x).
$$
Rewriting $y' = \dfrac{dy}{dx}$, you can think informally as
$$
\frac{dy}{dx} = f(x)
\quad\Rightarrow\quad
dy = f(x)\,dx,
$$
and then
$$
\int dy = \int f(x)\,dx
\quad\Rightarrow\quad
y = \int f(x)\,dx = F(x) + C.
$$

Thus, the set of all solutions is described by an indefinite integral.

If an initial condition is given (for example, $y(x_0) = y_0$), you can determine the particular constant $C$ that fits that condition. This use of indefinite integrals in solving differential equations is foundational and appears throughout calculus and beyond.

Common Pitfalls with Indefinite Integrals

Because indefinite integration reverses differentiation, many mistakes arise from forgetting how derivatives behave.

Some important points:

• Always include $+C$. Missing the constant of integration is a conceptual error (and important in many problems, especially differential equations).
• Do not treat $\int f(x)\,dx$ as a number; it is a function (or, more precisely, a family of functions).
• Linearity holds, but there is no simple general rule for $\int f(x)g(x)\,dx$ or $\int \frac{f(x)}{g(x)}\,dx$: products and quotients usually require techniques like substitution, integration by parts, or more advanced methods.
• Checking your work by differentiating your answer is usually the easiest way to catch algebraic mistakes.

Summary

Indefinite integrals:

• Represent families of antiderivatives of a given function.
• Use the notation $\displaystyle \int f(x)\,dx = F(x) + C$.
• Invert standard derivative formulas (power, exponential, logarithmic, trigonometric).
• Are linear: integrals of sums and constant multiples are straightforward.
• Rely on key techniques—substitution, integration by parts, trigonometric methods, and partial fractions—to handle more complicated integrands.
• Naturally describe general solutions to simple differential equations, with the constant of integration capturing “extra information” like initial conditions.

In later sections, definite integrals and the Fundamental Theorem of Calculus will connect these antiderivatives to areas and accumulation, but the indefinite integral itself is primarily about recovering a function from its derivative.