Indefinite Integrals for MATH 141

Exam Relevance for MATH 141

Likelihood of appearing: Medium

Indefinite integrals in MATH 141 are usually part of technique problems rather than standalone questions.

This skill appears on:

fall 2021 Final

Lesson

What is an Indefinite Integral?

An indefinite integral finds a function whose derivative gives you the original function — without evaluating at any specific points. It's also called finding an antiderivative.

Think of derivatives and integrals as opposites:

Derivative: You have $f(x) = x^2$, and you find $f'(x) = 2x$
Integral: You have $2x$, and you ask "what function has this as its derivative?" Answer: $x^2$

Integration is like running the derivative "backwards."

Key Notation

The indefinite integral of $f(x)$ is written as:

$$\int f(x) \, dx$$

Where:

$\int$ — the integral sign (an elongated "S" for "sum")
$f(x)$ — the integrand (the function you're integrating)
$dx$ — tells you the variable of integration

The Constant of Integration

Here's something crucial: if $F(x) = x^2$ has derivative $F'(x) = 2x$, then so does:

$G(x) = x^2 + 5$ → $G'(x) = 2x$
$H(x) = x^2 - 100$ → $H'(x) = 2x$
$J(x) = x^2 + \pi$ → $J'(x) = 2x$

Adding any constant doesn't change the derivative! So when we integrate, we must include + C (the constant of integration) to represent all possible antiderivatives:

$$\int 2x \, dx = x^2 + C$$

⚠️ Never forget the +C! It represents a whole family of functions that all have the same derivative.

Basic Integration Rules

These are the "reverse" of differentiation rules you already know.

Power Rule for Integration

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)}$$

Why? Because the derivative of $\frac{x^{n+1}}{n+1}$ is $x^n$.

Constant Multiple Rule

$$\int k \cdot f(x) \, dx = k \int f(x) \, dx$$

Constants can be "pulled out" of the integral.

Sum/Difference Rule

$$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$$

Integrate term by term.

Common Integrals to Memorize

Function	Integral
$\int k \, dx$	$kx + C$
$\int x^n \, dx$	$\frac{x^{n+1}}{n+1} + C$ (if $n \neq -1$)
$\int \frac{1}{x} \, dx$	$\ln\|x\| + C$
$\int e^x \, dx$	$e^x + C$
$\int \sin x \, dx$	$-\cos x + C$
$\int \cos x \, dx$	$\sin x + C$
$\int \sec^2 x \, dx$	$\tan x + C$

Example 1: Basic Power Rule

Find $\int x^4 \, dx$.

Using the power rule with $n = 4$:

$$\int x^4 \, dx = \frac{x^{4+1}}{4+1} + C = \frac{x^5}{5} + C$$

$$\boxed{\frac{x^5}{5} + C}$$

Example 2: Negative Exponent

Find $\int \frac{1}{x^3} \, dx$.

First, rewrite as a power: $\frac{1}{x^3} = x^{-3}$

Now apply the power rule with $n = -3$:

$$\int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} + C = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$$

$$\boxed{-\frac{1}{2x^2} + C}$$

Example 3: Polynomial Integration

Find $\int (3x^2 - 5x + 2) \, dx$.

Integrate term by term:

$$\int (3x^2 - 5x + 2) \, dx = \int 3x^2 \, dx - \int 5x \, dx + \int 2 \, dx$$

$$= 3 \cdot \frac{x^3}{3} - 5 \cdot \frac{x^2}{2} + 2x + C$$

$$= x^3 - \frac{5x^2}{2} + 2x + C$$

$$\boxed{x^3 - \frac{5x^2}{2} + 2x + C}$$

Example 4: Roots and Radicals

Find $\int \sqrt{x} \, dx$.

Rewrite the square root as a power: $\sqrt{x} = x^{1/2}$

Apply the power rule with $n = \frac{1}{2}$:

$$\int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C$$

Dividing by $\frac{3}{2}$ is the same as multiplying by $\frac{2}{3}$:

$$= \frac{2}{3}x^{3/2} + C = \frac{2}{3}\sqrt{x^3} + C$$

$$\boxed{\frac{2}{3}x^{3/2} + C}$$

Example 5: The Special Case $n = -1$

Find $\int \frac{1}{x} \, dx$.

The power rule doesn't work here (we'd get $\frac{x^0}{0}$, which is undefined).

This is a special case that gives us the natural logarithm:

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

We use absolute value $|x|$ because $\ln$ is only defined for positive numbers, but $\frac{1}{x}$ exists for negative $x$ too.

$$\boxed{\ln|x| + C}$$

Example 6: Exponential Function

Find $\int 5e^x \, dx$.

Pull out the constant and use the fact that $e^x$ is its own antiderivative:

$$\int 5e^x \, dx = 5 \int e^x \, dx = 5e^x + C$$

$$\boxed{5e^x + C}$$

Example 7: Trig Functions

Find $\int (2\sin x + 3\cos x) \, dx$.

Integrate term by term using the trig integral formulas:

$$\int (2\sin x + 3\cos x) \, dx = 2\int \sin x \, dx + 3\int \cos x \, dx$$

$$= 2(-\cos x) + 3(\sin x) + C$$

$$= -2\cos x + 3\sin x + C$$

$$\boxed{-2\cos x + 3\sin x + C}$$

Example 8: True or False

True or False: $\int x^2 \, dx = \frac{x^3}{3}$

False. The answer is missing the constant of integration. The correct answer is:

$$\int x^2 \, dx = \frac{x^3}{3} + C$$

Every indefinite integral must include $+ C$ because there are infinitely many functions with the same derivative (they differ by a constant).

Example 9: True or False

True or False: If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$, then $F(x) - G(x)$ must be a constant.

True. If $F'(x) = f(x)$ and $G'(x) = f(x)$, then:

$$\frac{d}{dx}[F(x) - G(x)] = F'(x) - G'(x) = f(x) - f(x) = 0$$

A function with zero derivative everywhere must be constant. So $F(x) - G(x) = C$ for some constant $C$.

This is why we write $\int f(x) \, dx = F(x) + C$ — all antiderivatives differ by at most a constant!

Example 10: True or False

True or False: $\int \frac{1}{x^2} \, dx = \ln|x^2| + C$

False. This confuses the special case $\int \frac{1}{x} \, dx = \ln|x| + C$ with the general power rule.

For $\frac{1}{x^2} = x^{-2}$, we use the power rule:

$$\int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$$

The $\ln$ formula only applies when the exponent is exactly $-1$.

Common Mistakes and Misunderstandings

❌ Mistake: Forgetting the +C

Wrong: $\int 2x \, dx = x^2$

Why it's wrong: Without the constant, you're only giving ONE antiderivative. The function $x^2 + 7$ also has derivative $2x$.

Correct: $\int 2x \, dx = x^2 + C$

❌ Mistake: Using the power rule when $n = -1$

Wrong: $\int \frac{1}{x} \, dx = \frac{x^0}{0} + C$ (undefined!)

Why it's wrong: The power rule $\frac{x^{n+1}}{n+1}$ fails when $n = -1$ because you'd divide by zero.

Correct: $\int \frac{1}{x} \, dx = \ln|x| + C$ (memorize this special case!)

❌ Mistake: Wrong sign for trig integrals

Wrong: $\int \sin x \, dx = \cos x + C$

Why it's wrong: The derivative of $\cos x$ is $-\sin x$, not $\sin x$.

Correct: $\int \sin x \, dx = -\cos x + C$

Check: $\frac{d}{dx}(-\cos x) = -(-\sin x) = \sin x$ ✓

❌ Mistake: Confusing integration with differentiation

Wrong: $\int x^3 \, dx = 3x^2 + C$

Why it's wrong: That's the derivative of $x^3$, not the integral! Integration is the reverse process.

Correct: $\int x^3 \, dx = \frac{x^4}{4} + C$

Check: $\frac{d}{dx}\left(\frac{x^4}{4}\right) = \frac{4x^3}{4} = x^3$ ✓

Formulas & Reference

Power Rule for Integration

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

The fundamental rule for integrating powers of x. Add 1 to the exponent, then divide by the new exponent. Does NOT work when n = -1 (use ln|x| instead).