Common Integration Formulas Dataset
This dataset provides a comprehensive list of 56 common integration formulas, detailing functions, their integrals, conditions, and notes. Ideal for students, educators, and professionals in mathematics.
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Key Takeaways
- Access 56 essential integration formulas for calculus.
- Explore categories like basic, trigonometric, and exponential functions.
- Download ready-to-use formulas with conditions and notes.
- Leverage accurate integral definitions for educational or professional use.
Showing 56 of 56
| Category | Function f(x) | ∫f(x)dx | Condition | Notes |
|---|---|---|---|---|
| Basic | k | kx + C | k is constant | Constant rule |
| Basic | x^n | x^(n+1)/(n+1) + C | n ≠ -1 | Power rule |
| Basic | 1/x | ln|x| + C | x ≠ 0 | Natural logarithm |
| Basic | e^x | e^x + C | - | Exponential function |
| Basic | a^x | a^x/ln(a) + C | a > 0, a ≠ 1 | General exponential |
| Basic | x^(-1/2) | 2√x + C | x > 0 | Square root form |
| Trigonometric | sin(x) | -cos(x) + C | - | Sine function |
| Trigonometric | cos(x) | sin(x) + C | - | Cosine function |
| Trigonometric | tan(x) | -ln|cos(x)| + C | x ≠ π/2 + nπ | Tangent function |
| Trigonometric | cot(x) | ln|sin(x)| + C | x ≠ nπ | Cotangent function |
| Trigonometric | sec(x) | ln|sec(x) + tan(x)| + C | x ≠ π/2 + nπ | Secant function |
| Trigonometric | csc(x) | ln|csc(x) - cot(x)| + C | x ≠ nπ | Cosecant function |
| Trigonometric | sec²(x) | tan(x) + C | x ≠ π/2 + nπ | Secant squared |
| Trigonometric | csc²(x) | -cot(x) + C | x ≠ nπ | Cosecant squared |
| Trigonometric | sec(x)tan(x) | sec(x) + C | x ≠ π/2 + nπ | Product form |
| Trigonometric | csc(x)cot(x) | -csc(x) + C | x ≠ nπ | Product form |
| Trigonometric | sin²(x) | x/2 - sin(2x)/4 + C | - | Half-angle identity |
| Trigonometric | cos²(x) | x/2 + sin(2x)/4 + C | - | Half-angle identity |
| Inverse Trig | 1/√(1-x²) | arcsin(x) + C | |x| < 1 | Arcsine form |
| Inverse Trig | -1/√(1-x²) | arccos(x) + C | |x| < 1 | Arccosine form |
| Inverse Trig | 1/(1+x²) | arctan(x) + C | - | Arctangent form |
| Inverse Trig | -1/(1+x²) | arccot(x) + C | - | Arccotangent form |
| Inverse Trig | 1/(|x|√(x²-1)) | arcsec(x) + C | |x| > 1 | Arcsecant form |
| Inverse Trig | -1/(|x|√(x²-1)) | arccsc(x) + C | |x| > 1 | Arccosecant form |
| Inverse Trig | 1/√(a²-x²) | arcsin(x/a) + C | |x| < a | General arcsine |
| Inverse Trig | 1/(a²+x²) | (1/a)arctan(x/a) + C | a ≠ 0 | General arctangent |
| Hyperbolic | sinh(x) | cosh(x) + C | - | Hyperbolic sine |
| Hyperbolic | cosh(x) | sinh(x) + C | - | Hyperbolic cosine |
| Hyperbolic | tanh(x) | ln(cosh(x)) + C | - | Hyperbolic tangent |
| Hyperbolic | coth(x) | ln|sinh(x)| + C | x ≠ 0 | Hyperbolic cotangent |
| Hyperbolic | sech(x) | arctan(sinh(x)) + C | - | Hyperbolic secant |
| Hyperbolic | csch(x) | ln|tanh(x/2)| + C | x ≠ 0 | Hyperbolic cosecant |
| Hyperbolic | sech²(x) | tanh(x) + C | - | Hyperbolic secant squared |
| Hyperbolic | csch²(x) | -coth(x) + C | x ≠ 0 | Hyperbolic cosecant squared |
| Logarithmic | ln(x) | x·ln(x) - x + C | x > 0 | Integration by parts |
| Logarithmic | log_a(x) | x·log_a(x) - x/ln(a) + C | x > 0, a > 0 | General logarithm |
| Logarithmic | 1/(x·ln(x)) | ln|ln(x)| + C | x > 1 | Nested logarithm |
| Rational | 1/(x²-a²) | (1/2a)ln|(x-a)/(x+a)| + C | x ≠ ±a | Partial fractions |
| Rational | 1/(a²-x²) | (1/2a)ln|(a+x)/(a-x)| + C | |x| < a | Partial fractions |
| Rational | 1/√(x²+a²) | ln|x + √(x²+a²)| + C | - | Hyperbolic substitution |
| Rational | 1/√(x²-a²) | ln|x + √(x²-a²)| + C | |x| > a | Hyperbolic substitution |
| Rational | 1/√(a²-x²) | arcsin(x/a) + C | |x| < a | Trigonometric substitution |
| Rational | √(a²-x²) | (x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C | |x| ≤ a | Trigonometric substitution |
| Rational | √(x²+a²) | (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C | - | Hyperbolic substitution |
| Rational | √(x²-a²) | (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C | |x| > a | Hyperbolic substitution |
| Special | e^(ax)sin(bx) | e^(ax)(a·sin(bx)-b·cos(bx))/(a²+b²) + C | - | Integration by parts twice |
| Special | e^(ax)cos(bx) | e^(ax)(a·cos(bx)+b·sin(bx))/(a²+b²) + C | - | Integration by parts twice |
| Special | x·e^x | e^x(x-1) + C | - | Integration by parts |
| Special | x·sin(x) | sin(x) - x·cos(x) + C | - | Integration by parts |
| Special | x·cos(x) | cos(x) + x·sin(x) + C | - | Integration by parts |
| Special | x²·e^x | e^x(x²-2x+2) + C | - | Integration by parts twice |
| Special | x·ln(x) | (x²/2)ln(x) - x²/4 + C | x > 0 | Integration by parts |
| Definite | ∫₀^∞ e^(-x²) dx | √π/2 | - | Gaussian integral |
| Definite | ∫₀^∞ x^n·e^(-x) dx | n! (Gamma function) | n ≥ 0 integer | Gamma function relation |
| Definite | ∫₀^π sin^n(x) dx | Wallis formula | n ≥ 0 integer | Reduction formula |
| Definite | ∫₀^(π/2) sin^n(x)cos^m(x) dx | Beta function | n,m > -1 | Beta function relation |
Use Cases
- Import the CSV file into your Python script or SQL database to build custom calculus solvers or educational tools.
- Use the Excel file to filter formulas by category, create study guides, or verify calculations for complex problems.
- Print the PDF version for a quick reference guide in classrooms, during exams, or for offline study sessions.
- Integrate this dataset into e-learning platforms to provide a structured resource for students learning integral calculus.