Lists of integrals

Lists of integrals typically refer to collections or tables that provide the integrals of various functions, which can be useful for students and mathematicians when solving calculus problems. These lists usually include both definite and indefinite integrals, covering a wide range of functions, including polynomial, trigonometric, exponential, logarithmic, and special functions. The format of a list of integrals will often present the integral alongside its result, often accompanied by conditions related to the variables in the integrals.

The list of integrals of Gaussian functions includes several important results involving integrals of the form \[ I(a, b) = \int_{-\infty}^{\infty} e^{-ax^2 + bx} \, dx \] where \( a > 0 \) and \( b \) is a constant.

The integral of exponential functions is a fundamental topic in calculus. Here’s a list of some common integrals involving exponential functions: 1. **Basic Exponential Function**: \[ \int e^x \, dx = e^x + C \] 2.

The integrals of hyperbolic functions are useful in various fields such as calculus, physics, and engineering. Here is a list of some common integrals involving hyperbolic functions: 1. **Basic Hyperbolic Functions:** - \(\int \sinh(x) \, dx = \cosh(x) + C\) - \(\int \cosh(x) \, dx = \sinh(x) + C\) 2.

The integrals of inverse trigonometric functions are commonly encountered in calculus. Below is a list of the integrals for the six primary inverse trigonometric functions: 1. **Integral of arcsin(x)**: \[ \int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1 - x^2} + C \] 2.

The term "list of integrals of irrational functions" typically refers to a collection of integrals that involve irrational functions—functions which cannot be expressed as a ratio of polynomials. This includes functions containing roots, such as square roots, cube roots, and other higher-degree roots, as well as logarithmic and exponential functions that may have irrational components.

The integral of logarithmic functions is a common topic in calculus. Here’s a list of some common integrals involving logarithmic functions: 1. **Integral of ln(x)**: \[ \int \ln(x) \, dx = x \ln(x) - x + C \] 2.

The list of integrals of rational functions consists of formulas that help integrate rational functions, which are functions that can be expressed as the ratio of two polynomials. The general form of a rational function is \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. To integrate rational functions, a common approach is to use polynomial long division followed by partial fraction decomposition.

Integrals of trigonometric functions are frequently encountered in calculus. Below is a list of common integrals involving the basic trigonometric functions: ### Basic Trigonometric Integrals 1. **Sine Function:** \[ \int \sin(x) \, dx = -\cos(x) + C \] 2.