Main Integrals- Essential Integration Formulas

What Are Main Integrals and Why You Need to Know Them

Integrals are the backbone of calculus. If you can't solve basic integrals, you'll struggle with anything beyond introductory math—physics, engineering, statistics, even machine learning all depend on integration.

The main integrals are the building blocks. Memorize these, practice them until they're automatic, and you'll handle most problems that come your way.

Basic Integral Formulas You Must Know

These are the foundation. No exceptions.

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C — Works for any n except -1
• ∫ 1/x dx = ln|x| + C — The one exception to the power rule
• ∫ eˣ dx = eˣ + C — Exponential stays the same
• ∫ aˣ dx = aˣ/ln(a) + C — Add the logarithm denominator
• ∫ sin(x) dx = -cos(x) + C — Watch the sign
• ∫ cos(x) dx = sin(x) + C — This one's cleaner
• ∫ sec²(x) dx = tan(x) + C
• ∫ csc²(x) dx = -cot(x) + C

Trigonometric Integrals Worth Memorizing

These come up constantly in physics and engineering problems.

• ∫ tan(x) dx = -ln|cos(x)| + C = ln|sec(x)| + C
• ∫ cot(x) dx = ln|sin(x)| + C
• ∫ sec(x) dx = ln|sec(x) + tan(x)| + C
• ∫ csc(x) dx = -ln|csc(x) + cot(x)| + C

The logarithmic forms look ugly, but they appear everywhere. Don't skip them.

Hyperbolic Function Integrals

Less common, but you'll hit these if you're doing electrical engineering or advanced physics.

• ∫ sinh(x) dx = cosh(x) + C
• ∫ cosh(x) dx = sinh(x) + C
• ∫ sech²(x) dx = tanh(x) + C

Integration Methods Compared

Sometimes you need more than the basic formulas. Here's when to use each technique:

How to Get Started: Substitution

Substitution is your first line of attack. Here's the process:

1. Pick a part of the integrand to set as u
2. Find du by differentiating
3. Replace everything in terms of u
4. Integrate with respect to u
5. Substitute x back in

Example: ∫ 2x·cos(x²) dx

Let u = x², so du = 2x dx

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

Done. That's it.

Integration by Parts: When Substitution Fails

Use this when you have a product of a polynomial and something else.

The formula: ∫ u dv = uv - ∫ v du

Example: ∫ x·eˣ dx

Let u = x (so du = dx), dv = eˣ dx (so v = eˣ)

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

Pick u using LIATE: Log, Inverse trig, Algebraic, Trigonometric, Exponential. Higher on the list gets u.

Definite Integrals: Adding the Bounds

Definite integrals need the Fundamental Theorem of Calculus.

∫ₐᵇ f(x) dx = F(b) - F(a)

Find the antiderivative, plug in the bounds, subtract.

Example: ∫₀² x² dx

= [x³/3]₀² = (8/3) - (0) = 8/3

Common Mistakes to Avoid

• Forgetting the +C on indefinite integrals
• Getting the sign wrong on trigonometric integrals
• Trying to use substitution when integration by parts is needed
• Dropping absolute values in logarithm answers
• Solving when you should be integrating

Practice Problems to Master These Formulas

Work through these. Check your answers.

1. ∫ x⁵ dx
2. ∫ 3x²·sin(x³) dx
3. ∫ sec(x)tan(x) dx
4. ∫ x·ln(x) dx
5. ∫₁ᵉ 1/x dx

Answers: x⁶/6 + C | -cos(x³) + C | sec(x) + C | x²·ln(x)/2 - x²/4 + C | 1

Final Thoughts

You don't need to understand integration deeply to use these formulas. Practice until you recognize patterns. Most integrals you'll encounter are variations of these basics.

When you're stuck, try substitution first. If that fails, try integration by parts. If that fails too, you probably need partial fractions or trig substitution—go back to the table above.

That's it. Now go practice.