Integral Rules- Essential Formulas for Integration

What Integration Actually Is

Integration is finding the area under a curve. That's it. No philosophical nonsense. You have a function, you want the total accumulation of that function over an interval, integration gives you that number.

Differentiation breaks things apart. Integration sticks them back together. They're inverse operations, which means if you know your derivative rules, you're halfway to knowing integration.

Basic Integration Rules You Need to Memorize

These are the foundation. Without these committed to memory, you're dead in the water.

The Power Rule

This is the one you'll use most often. For any exponent except -1:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C

Add one to the exponent, divide by the new exponent. Always add the constant C.

The Constant Multiple Rule

Constants slide out of the integral:

∫k·f(x) dx = k · ∫f(x) dx

If you have a 5 outside, the 5 stays outside while you integrate the function.

The Sum/Difference Rule

Integrate each term separately:

∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

The same applies for subtraction. Break it up, integrate each piece.

The Constant Rule

∫k dx = kx + C

Integrating a constant just multiplies it by x.

Essential Integration Formulas

Beyond the power rule, these come up constantly in problems:

• ∫dx/x = ln|x| + C — This is the one exception to the power rule. When you see x to the -1, you get a natural log, not a fraction.
• ∫eˣ dx = eˣ + C — e to the x integrates to itself. Beautiful in its simplicity.
• ∫eᵘ du = eᵘ + C — Same deal with substitutions.
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫sec²(x) dx = tan(x) + C
• ∫csc²(x) dx = -cot(x) + C
• ∫sec(x)tan(x) dx = sec(x) + C
• ∫1/(1+x²) dx = arctan(x) + C
• ∫1/√(1-x²) dx = arcsin(x) + C

Memorize the trig ones. They show up in substitution problems constantly.

Common Integration Formulas Compared

Integration Techniques for Harder Problems

Basic formulas only get you so far. When the function looks ugly, you need these techniques.

U-Substitution

This is the first technique you learn and the one you'll use most. It's the chain rule in reverse.

How it works:

1. Pick a part of the integrand to call "u"
2. Find du by taking the derivative of u
3. Substitute everything — u for the chosen part, du for its derivative
4. Integrate in terms of u
5. Substitute x back in for u

Example: ∫2x·cos(x²) dx

Let u = x², then du = 2x dx

The integral becomes ∫cos(u) du = sin(u) + C = sin(x²) + C

Integration by Parts

For products of functions. It's the product rule in reverse.

Formula: ∫u dv = uv - ∫v du

Choosing u: Use LIATE — pick the function higher on the list as u:

• Logarithmic
• Inverse trig
• Algebraic (polynomials)
• Trigonometric
• Exponential

Example: ∫x·eˣ dx

Let u = x, dv = eˣ dx

Then du = dx, v = eˣ

∫x·eˣ dx = x·eˣ - ∫eˣ dx = x·eˣ - eˣ + C = eˣ(x-1) + C

Trigonometric Substitution

Use this when you see square roots with quadratic expressions inside.

After substituting, simplify using trig identities, integrate, then convert back to x using right triangles.

Getting Started: Solving Your First Integral

Here's the process for tackling any integration problem:

1. Identify the type. Is it a basic formula, a substitution, parts, or trig sub?
2. Check for obvious substitutions. Look for something inside a composite function — that's usually your u.
3. Look for products. If you see two different function types multiplied, consider integration by parts.
4. Simplify first. Distribute, split fractions, rewrite — algebra before you integrate.
5. Don't forget +C. Every indefinite integral needs the constant.

Work through ∫(3x² + 2x - 5) dx step by step:

Split it: ∫3x² dx + ∫2x dx - ∫5 dx

Power rule on each: x³ + x² - 5x + C

That's it. That's the answer.

Common Mistakes to Avoid

• Forgetting the constant. Every indefinite integral needs +C. Always.
• Wrong sign on the power rule. ∫x⁻¹ dx = ln|x|, not x⁰/0. The power rule doesn't apply here.
• Integration is not differentiation backwards. ∫f'(x) dx = f(x) + C, not f'(x). The function comes back, not its derivative.
• Substitution errors. If du doesn't appear in your integral after substituting, you picked the wrong u.
• Parts sign errors. The formula is uv - ∫v du. The minus sign stays. People mess this up constantly.

When to Use What

Quick decision tree:

• Simple power of x? → Power rule
• 1/x? → ln|x|
• eˣ or eᵘ? → Same function back
• sin or cos? → Other trig function with sign flip
• Composite function? → u-substitution
• Product of different function types? → Integration by parts
• Square root of quadratic? → Trig substitution

That's the full picture. These rules and formulas cover the vast majority of integration problems you'll encounter. Practice the techniques until the process becomes automatic.