∫ Indefinite Integrals and Notation

The Integral Symbol

The process of finding antiderivatives is called integration, and we use the integral symbol:

$\int f(x)\,dx$

This is read as "the integral of $f(x)$ with respect to $x$."

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Notation Breakdown

$\int f(x)\,dx = F(x) + C$

Components:

• $\int$ - integral symbol (looks like an elongated S for "sum")
• $f(x)$ - the integrand (function being integrated)
• $dx$ - tells us we're integrating with respect to $x$
• $F(x) + C$ - the antiderivative (general solution)

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Indefinite vs. Definite Integrals

Indefinite Integral

$\int f(x)\,dx = F(x) + C$

• Represents the family of all antiderivatives
• Includes the constant $+C$
• Result is a function

Definite Integral (coming later!)

$\int_a^b f(x)\,dx = F(b) - F(a)$

• Has limits of integration ($a$ to $b$)
• Represents a specific number (area)
• No $+C$ needed

For now, we focus on indefinite integrals!

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Basic Integration Formulas

Power Rule

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

Examples:

• $\int x^5\,dx = \frac{x^6}{6} + C$
• $\int x\,dx = \frac{x^2}{2} + C$
• $\int 1\,dx = x + C$

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Special Case: $n = -1$

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

Why absolute value?

• $\ln x$ only defined for $x > 0$
• $\ln|x|$ works for all $x \neq 0$

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Constant Rule

$\int k\,dx = kx + C$

where $k$ is any constant.

Example: $\int 5\,dx = 5x + C$

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Exponential Function

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

The exponential function is its own integral!

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Trigonometric Functions

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

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

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

$\int \sec x \tan x\,dx = \sec x + C$

$\int \csc^2 x\,dx = -\cot x + C$

$\int \csc x \cot x\,dx = -\csc x + C$

Remember: Sine → negative cosine, Cosine → positive sine

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Properties of Integrals

Constant Multiple Rule

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

You can "pull out" constants!

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

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Sum/Difference Rule

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

Integrate term by term!

Example: $\int (x^2 + 3x)\,dx = \int x^2\,dx + \int 3x\,dx$ $= \frac{x^3}{3} + \frac{3x^2}{2} + C$

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Important: The $dx$

The "$dx$" is not optional - it tells us the variable of integration!

Example: $\int t^2\,dt$ vs $\int x^2\,dx$

Same form, different variables!

Why it matters:

• Clarifies which variable we're integrating
• Essential for u-substitution (later topic)
• Part of the mathematical notation

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Working with Integrals

Example 1: Polynomial

Evaluate $\int (4x^3 - 2x^2 + 5)\,dx$

Solution:

Apply sum rule and constant multiple rule:

$= \int 4x^3\,dx - \int 2x^2\,dx + \int 5\,dx$

$= 4\int x^3\,dx - 2\int x^2\,dx + \int 5\,dx$

$= 4 \cdot \frac{x^4}{4} - 2 \cdot \frac{x^3}{3} + 5x + C$

$= x^4 - \frac{2x^3}{3} + 5x + C$

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Example 2: Rewriting Before Integrating

Evaluate $\int \frac{3}{x^4}\,dx$

Step 1: Rewrite using exponents

$\int \frac{3}{x^4}\,dx = \int 3x^{-4}\,dx$

Step 2: Apply power rule

$= 3 \cdot \frac{x^{-4+1}}{-4+1} + C = 3 \cdot \frac{x^{-3}}{-3} + C$

$= -\frac{3}{3x^3} + C = -\frac{1}{x^3} + C$

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Example 3: Expanding First

Evaluate $\int (x+1)^2\,dx$

WRONG approach: Try to integrate directly ❌

RIGHT approach: Expand first!

$\int (x+1)^2\,dx = \int (x^2 + 2x + 1)\,dx$

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

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

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Example 4: Trigonometric

Evaluate $\int (3\sin x - 2\cos x)\,dx$

Solution:

$= 3\int \sin x\,dx - 2\int \cos x\,dx$

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

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

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Fractional and Negative Exponents

Square Roots

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

Remember: $\sqrt{x} = x^{1/2}$

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Cube Roots

$\int \sqrt[3]{x^2}\,dx = \int x^{2/3}\,dx = \frac{x^{5/3}}{5/3} + C = \frac{3x^{5/3}}{5} + C$

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Reciprocals

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

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Combining Multiple Techniques

Example: Mixed Terms

Evaluate $\int \left(x^2 + \frac{2}{x^2} + \sqrt{x}\right)dx$

Step 1: Rewrite everything as powers

$= \int \left(x^2 + 2x^{-2} + x^{1/2}\right)dx$

Step 2: Integrate term by term

$= \frac{x^3}{3} + 2 \cdot \frac{x^{-1}}{-1} + \frac{x^{3/2}}{3/2} + C$

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

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The "$+C$" Convention

When combining multiple constants, we can consolidate:

$\int (x^2 + 1)\,dx = \frac{x^3}{3} + C_1 + x + C_2$

We typically write this as: $= \frac{x^3}{3} + x + C$

where $C = C_1 + C_2$ (arbitrary constant).

One +C is enough at the end!

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⚠️ Common Mistakes

Mistake 1: Forgetting +C

WRONG: $\int x^2\,dx = \frac{x^3}{3}$

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

Always include the constant of integration!

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Mistake 2: Wrong Power Rule

WRONG: $\int x^3\,dx = \frac{x^3}{3} + C$ (forgot to add 1 to exponent)

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

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Mistake 3: Forgetting $dx$

WRONG: $\int x^2$

RIGHT: $\int x^2\,dx$

The $dx$ is part of the notation!

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Mistake 4: Can't Integrate Products/Quotients Directly

WRONG: $\int (x \cdot x^2)\,dx = \left(\int x\,dx\right) \cdot \left(\int x^2\,dx\right)$

RIGHT: Simplify first! $\int x^3\,dx = \frac{x^4}{4} + C$

There's no "product rule" for integrals!

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Mistake 5: Sine/Cosine Signs

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

RIGHT: $\int \sin x\,dx = -\cos x + C$ (negative!)

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When Basic Rules Don't Work

Some integrals need advanced techniques:

Can't do easily:

• $\int e^{x^2}\,dx$ (needs special functions)
• $\int \sin(x^2)\,dx$ (needs special functions)
• $\int \frac{1}{x^2+1}\,dx$ (this is $\arctan x + C$, learned later)

Can do with techniques:

• $\int x\cos(x^2)\,dx$ (u-substitution)
• $\int x e^x\,dx$ (integration by parts)

We'll learn these methods in future lessons!

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Quick Reference Table

| Integrand | Integral | |-----------|----------| | $x^n$ ($n \neq -1$) | $\frac{x^{n+1}}{n+1} + C$ | | $\frac{1}{x}$ | $\ln|x| + C$ | | $e^x$ | $e^x + C$ | | $\sin x$ | $-\cos x + C$ | | $\cos x$ | $\sin x + C$ | | $\sec^2 x$ | $\tan x + C$ | | $\sec x \tan x$ | $\sec x + C$ |

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📝 Practice Strategy

1. Rewrite the integrand if needed (expand, use exponents)
2. Apply linearity (split sums, pull out constants)
3. Use basic formulas (power rule, trig, exponential)
4. Always include +C
5. Check your answer by differentiating
6. Include $dx$ in your notation
7. Simplify before integrating when possible