Integration Formulas
Here’s the deal: integration formulas are the foundation of calculus. You can derive them from scratch every time, or you can memorize the essential ones and actually get work done. I’ve spent years teaching this stuff, and these are the formulas that show up constantly.
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You can read it online or download the integration formulas PDF here.
Basic Integration Rules
These are your bread and butter. Master these first—everything else builds on them.
| Rule | Formula | Condition |
|---|---|---|
| Power Rule | $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ | \( n \neq -1 \) |
| Constant | $$\int a \, dx = ax + C$$ | — |
| Constant Multiple | $$\int k \cdot f(x) \, dx = k \int f(x) \, dx$$ | — |
| Sum/Difference | $$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$$ | — |
Pro Tip: The Power Rule handles about 60% of the integrals you’ll encounter. When \( n = -1 \), you get the natural log (see below). Don’t forget that \( C \)—indefinite integrals always have a constant of integration.
Exponential and Logarithmic Functions
Exponentials and logs are everywhere—growth models, decay problems, compound interest. Here’s what you need:
- Natural Exponential: $$\int e^x \, dx = e^x + C$$
- General Exponential: $$\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$$
- Base \( a \) Exponential: $$\int a^x \, dx = \frac{a^x}{\ln a} + C, \quad a > 0, \, a \neq 1$$
- Natural Logarithm (from 1/x): $$\int \frac{1}{x} \, dx = \ln|x| + C$$
- Logarithm Integration: $$\int \ln x \, dx = x \ln x – x + C$$
- Product of \( x \) and Exponential: $$\int x e^x \, dx = e^x(x – 1) + C$$
The absolute value in \( \ln|x| \) matters. Without it, the formula only works for positive \( x \). With it, you’re covered for all \( x \neq 0 \).
Trigonometric Function Integrals
Trig integrals are the workhorses of physics and engineering. I’ve organized these by how often you’ll actually use them.
Core Trig Integrals (Use Daily)
| Function | Integral |
|---|---|
| \( \sin x \) | $$\int \sin x \, dx = -\cos x + C$$ |
| \( \cos x \) | $$\int \cos x \, dx = \sin x + C$$ |
| \( \sec^2 x \) | $$\int \sec^2 x \, dx = \tan x + C$$ |
| \( \csc^2 x \) | $$\int \csc^2 x \, dx = -\cot x + C$$ |
| \( \sec x \tan x \) | $$\int \sec x \tan x \, dx = \sec x + C$$ |
| \( \csc x \cot x \) | $$\int \csc x \cot x \, dx = -\csc x + C$$ |
Advanced Trig Integrals
These take longer to derive but show up in real problems:
- $$\int \sec x \, dx = \ln|\sec x + \tan x| + C$$
- $$\int \csc x \, dx = \ln|\csc x – \cot x| + C$$
- $$\int \tan x \, dx = -\ln|\cos x| + C = \ln|\sec x| + C$$
- $$\int \cot x \, dx = \ln|\sin x| + C$$
Squared Trig Functions
These come from half-angle identities. Memorize them or derive them—your call:
- $$\int \sin^2 x \, dx = \frac{x}{2} – \frac{\sin 2x}{4} + C$$
- $$\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$$
- $$\int \sin x \cos x \, dx = \frac{\sin^2 x}{2} + C$$
Inverse Trigonometric Function Integrals
When you see expressions like \( \frac{1}{\sqrt{1-x^2}} \), think inverse trig. These patterns are unmistakable once you learn to spot them.
| Integrand | Result | Domain |
|---|---|---|
| \( \displaystyle\frac{1}{\sqrt{1-x^2}} \) | $$\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1} x + C$$ | \( |x| < 1 \) |
| \( \displaystyle\frac{1}{1+x^2} \) | $$\int \frac{1}{1+x^2} \, dx = \tan^{-1} x + C$$ | All \( x \) |
| \( \displaystyle\frac{1}{x\sqrt{x^2-1}} \) | $$\int \frac{1}{x\sqrt{x^2-1}} \, dx = \sec^{-1} x + C$$ | \( |x| > 1 \) |
| \( \displaystyle\frac{-1}{\sqrt{1-x^2}} \) | $$\int \frac{-1}{\sqrt{1-x^2}} \, dx = \cos^{-1} x + C$$ | \( |x| < 1 \) |
| \( \displaystyle\frac{-1}{1+x^2} \) | $$\int \frac{-1}{1+x^2} \, dx = \cot^{-1} x + C$$ | All \( x \) |
Pattern Recognition: Square root in denominator with \( 1 – x^2 \)? Think arcsine. Just \( 1 + x^2 \) in denominator? Think arctangent. This pattern extends to the general forms below.
Hyperbolic Function Integrals
Hyperbolic functions mirror their trig counterparts, but the signs are friendlier:
- $$\int \sinh x \, dx = \cosh x + C$$
- $$\int \cosh x \, dx = \sinh x + C$$
- $$\int \text{sech}^2 x \, dx = \tanh x + C$$
- $$\int \text{csch}^2 x \, dx = -\coth x + C$$
- $$\int \tanh x \, dx = \ln(\cosh x) + C$$
- $$\int \coth x \, dx = \ln|\sinh x| + C$$
Notice: \( \int \sinh x \, dx = \cosh x \) (no negative sign) unlike \( \int \sin x \, dx = -\cos x \). That’s the beauty of hyperbolics.
Important Algebraic Integrals
These formulas handle rational functions and expressions with quadratics. They’re essential for partial fractions and completing the square.
Linear Substitution Forms
- Linear Power: $$\int (ax + b)^n \, dx = \frac{1}{a} \cdot \frac{(ax + b)^{n+1}}{n+1} + C, \quad n \neq -1$$
- Reciprocal Power: $$\int \frac{1}{(x+a)^n} \, dx = -\frac{1}{(n-1)(x+a)^{n-1}} + C, \quad n \neq 1$$
Quadratic Forms
These show up constantly in physics (especially with inverse-square laws):
- Arctangent Form: $$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a} + C$$
- Difference of Squares: $$\int \frac{1}{x^2 – a^2} \, dx = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C$$
- Square Root Sum: $$\int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \ln|x + \sqrt{x^2 + a^2}| + C$$
- Square Root Difference: $$\int \frac{1}{\sqrt{x^2 – a^2}} \, dx = \ln|x + \sqrt{x^2 – a^2}| + C$$
- Arcsine Form: $$\int \frac{1}{\sqrt{a^2 – x^2}} \, dx = \sin^{-1} \frac{x}{a} + C$$
Integration Techniques
When formulas alone don’t cut it, you need techniques. Here are the big three:
Integration by Parts
$$\int u \, dv = uv – \int v \, du$$
LIATE Rule for choosing \( u \): Logarithmic → Inverse trig → Algebraic → Trig → Exponential. Pick \( u \) from whatever comes first in this list.
U-Substitution
$$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x)$$
Look for a function and its derivative appearing together. That’s your signal to substitute.
Trigonometric Substitution
| Expression | Substitution | Identity Used |
|---|---|---|
| \( \sqrt{a^2 – x^2} \) | \( x = a \sin \theta \) | \( 1 – \sin^2 \theta = \cos^2 \theta \) |
| \( \sqrt{a^2 + x^2} \) | \( x = a \tan \theta \) | \( 1 + \tan^2 \theta = \sec^2 \theta \) |
| \( \sqrt{x^2 – a^2} \) | \( x = a \sec \theta \) | \( \sec^2 \theta – 1 = \tan^2 \theta \) |
Special Functions and Applications
These show up in advanced calculus and mathematical physics:
Beta and Gamma Functions
- Beta Function: $$B(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}$$
- Gamma Function: $$\Gamma(n) = \int_0^{\infty} x^{n-1} e^{-x} \, dx$$
Key fact: \( \Gamma(n) = (n-1)! \) for positive integers. The Gamma function extends factorials to non-integers.
Geometric Applications
- Arc Length: $$L = int_a^b sqrt{1 + left(frac{dy}{dx}right)^2} , dx$$
- Surface Area of Revolution (about x-axis): $$S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
- Volume of Revolution (Disk Method about x-axis): $$V = \int_a^b \pi y^2 \, dx$$
- Volume of Revolution (about y-axis): $$V = \int_c^d \pi x^2 \, dy$$
Quick Reference Card
Here’s everything in one place for quick lookup:
| Category | \( f(x) \) | \( \int f(x) \, dx \) |
|---|---|---|
| Power | \( x^n \) | \( \frac{x^{n+1}}{n+1} + C \) |
| Exponential | \( e^x \) | \( e^x + C \) |
| Logarithmic | \( \frac{1}{x} \) | \( \ln|x| + C \) |
| Sine | \( \sin x \) | \( -\cos x + C \) |
| Cosine | \( \cos x \) | \( \sin x + C \) |
| Secant squared | \( \sec^2 x \) | \( \tan x + C \) |
| Inverse sine | \( \frac{1}{\sqrt{1-x^2}} \) | \( \sin^{-1} x + C \) |
| Inverse tangent | \( \frac{1}{1+x^2} \) | \( \tan^{-1} x + C \) |
| Hyperbolic sine | \( \sinh x \) | \( \cosh x + C \) |
| Hyperbolic cosine | \( \cosh x \) | \( \sinh x + C \) |
Frequently Asked Questions
What is the constant of integration C?
The constant C represents all possible antiderivatives of a function. Since the derivative of any constant is zero, adding C to an indefinite integral accounts for all functions whose derivative equals the integrand. For definite integrals with limits, C cancels out and isn’t written.
Why does the power rule not work when n = -1?
When n = -1, the formula x^(n+1)/(n+1) becomes x^0/0 = 1/0, which is undefined. Instead, the integral of x^(-1) = 1/x is ln|x| + C. This is a special case that requires its own formula because logarithms are the antiderivatives of 1/x.
When should I use integration by parts vs. u-substitution?
Use u-substitution when you see a function and its derivative together (like x·e^(x²)). Use integration by parts when you have a product of different types of functions (like x·sin(x) or x·e^x). The LIATE rule helps choose which part to call u in integration by parts.
What is the difference between definite and indefinite integrals?
An indefinite integral has no limits and produces a family of functions (the antiderivative plus C). A definite integral has upper and lower limits and produces a specific number representing area under the curve. Use the same formulas for both, but evaluate definite integrals at the limits using the Fundamental Theorem of Calculus.
Why is there a negative sign in the integral of sin(x)?
The integral of sin(x) is -cos(x) because the derivative of -cos(x) is sin(x). Remember: d/dx[-cos(x)] = -(-sin(x)) = sin(x). The negative sign comes from the chain of derivatives between sine and cosine functions. You can verify any integral formula by differentiating the result.
How do I know which trigonometric substitution to use?
Match the pattern: For √(a² – x²), use x = a·sin(θ). For √(a² + x²), use x = a·tan(θ). For √(x² – a²), use x = a·sec(θ). Each substitution converts the square root into a single trig function using Pythagorean identities.
What is the LIATE rule for integration by parts?
LIATE is a mnemonic for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Choose u from whatever type appears first in this list. For example, in ∫x·e^x dx, x is algebraic and e^x is exponential, so u = x (algebraic comes before exponential).
Why do inverse trig functions appear in integration formulas?
Inverse trig functions are the antiderivatives of certain algebraic expressions. For example, d/dx[arctan(x)] = 1/(1+x²), so ∫1/(1+x²) dx = arctan(x) + C. These connections arise naturally from the derivatives of inverse trig functions and are not arbitrary.
What are hyperbolic functions and when do I use them?
Hyperbolic functions (sinh, cosh, tanh, etc.) are defined using exponentials: sinh(x) = (e^x – e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. They appear in physics (hanging cables, special relativity), engineering (signal processing), and whenever solutions involve exponential growth and decay combinations.
How do I verify that my integral is correct?
Differentiate your answer. If you get back the original integrand, your integral is correct. This works because integration and differentiation are inverse operations. For example, if you think ∫sin(x) dx = -cos(x) + C, verify by checking that d/dx[-cos(x)] = sin(x). Always verify when unsure.